LIMITING THEOREMS FOR STATIONARY AND ERGODIC QUANTUM PROCESSES By Lubashan Pathirana A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics—Doctor of Philosophy 2023 ABSTRACT A discrete time parameter quantum process is obtained by iterated compositions of quan- tum operations. A quantum operation, a generalization of a quantum channel, is a com- pletely positive map that is not necessarily trace preserving. As such, a discrete param- eter quantum process is described by a sequence of quantum operations. We study the cases where the sequence is strictly stationary. By allowing the quantum operations to not preserve the trace one may study cases such as moment generating functions for a measurement processes or even measurement processes that discard the system upon a certain measurement outcome. An ergodic theorem describing convergence to equilib- rium for the class of ergodic quantum processes was recently obtained by Movassagh and Schenker in [56, 55]. We derive similar theorems for the strictly stationary case. Further- more, under certain irreducubility and mixing conditions, we see that the assymptotics of such processes are governed by the top (maximal) Lyapunov exponent for the ergodic sequences and we derive a law of large numbers (LLN) and a central limit theorem (CLT). In the continuous time-parameter a quantum process is described through a doubled in- dexed family of quantum operations such that the compositions of maps preserve the dynamics. Here we generalize the results in [56, 55] for the continuous time-parameter and strictly stationary case. To Chamila without whom this thesis would have many typos & to Mishini without whom this thesis would have more pages. iii ACKNOWLEDGEMENTS I am greatly indebted to chair of my committee, my advisor, Prof. Jeffrey Schenker firstly for guiding me during a time in my PhD career that I mostly needed guidance and then for his patience and feedback during my research. I also would not be able to success- fully complete my PhD without my defense committee, I would like to thank Prof. Ilya Kachkovskiy, Prof. Alexander Volberg and Prof. Yimin Xiao for the courses they taught me, questions they asked me on various occasions and writing me recommendation let- ters. I also like to thank Prof. Peter Hislop (University of Kentucky) for writing me recom- mendation letters and for discussions on his research. I am also grateful for professors who have taught me intriguing concepts in math that I am using for my research and have used for the thesis: In particular, I would like to thank Prof. Dapeng Zhan for teaching me probability and stochastic analysis, Prof. Todd Young (Ohio University) for teaching me analysis. A PhD career will not be successful without also being a good instructor / Teaching assistant. I like to express my sincere gratitude for the people who helped me to become a successful teacher. I would like to thank Tsveta Sendova and Andrew Krause for the phenomenal guidance they offered me and my colleagues for us to become great college instructors and helping and guiding graduate students thought our PhD careers at MSU. I also like to thank my first teaching mentor at MSU, Irina Holmes (currently in Texas A & M) for mentoring me in my first year and Prof. Brent Nelson for his guidance during the time I was a grader for complex analysis. A special thank also goes to Laura Willoughby for the tremendous support she gives for the graduate students in the math department and I extend this thank-you to Estrella Starn for the similar role she played while in the math department. I would like to extend my sincere thanks for my colleagues, my friends, some of whom are still in MSU and some of whom who are already graduated, for many things such as: iv serving in committees with me, discussions in math, working and discussing on first year homework together, covering my teaching duties when I couldn’t, etc. The thank you goes to: Arman, Christoper, Craig, Danika, Eric (Cheuk), Franciska, Hitesh, Jacob, Joe, Kai, Keshav, Leo, Melina, Mike, Rachel, Reshma, Rithwik, Rodrigo, Samara, Shashini, Shika and Vindya. I also like to thank Ana-Maria, Dimitris, Eranda, Estefania, Mihalis, Manos, Sanjaya, and Stan (Suo-Jun) for mortal support and being great friends. I also like to thank my friends, Eloy and Owen, for their collaborations in our research. I take this opportunity to thank my friends from college: Aravinda, Buddhima, Chandika, Kasun and Prashanth, hopefully we may be able to get together sometime soon. Finally, I would like to thank my parents, my parents in-law and my wife. Their belief has kept my spirits up and has motivated me during this process. I would also like to thank my baby daughter for her love and her help in keeping this thesis brief. v TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Random Compositions of Quantum Channels . . . . . . . . . . . . . . . 2 1.1.1 Random repeated interactions . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Repeated Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Positivity Preserving Maps: A More General Class Than Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Organization of the Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 5 CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Probability and Ergodicity Preliminaries . . . . . . . . . . . . . . . . . . 6 2.2 Stationary Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Natural Extension of a Measure Preserving Transformation . . . . . . . 16 2.4 Matrices and Super Operators . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Quantum Channels and Quantum Operations . . . . . . . . . . . . . . . 24 2.6 Geometry on States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Positive Super Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Contraction Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CHAPTER 3 DISCRETE PARAMETER QUANTUM PROCESSES . . . . . . . . 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Stationary Quantum Processes . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Definition and Assumptions . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Limiting Results for the Stationary Case in Forward Time . . . . . . 48 3.2.3 Double-sided Stationary Processes: Definitions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.4 Limiting Results for Two-sided Stationary Processes . . . . . . . . . 53 3.3 Discrete Parameter Ergodic Quantum Processes . . . . . . . . . . . . . . 62 3.3.1 Definition and Assumption . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.2 Limiting Results in Forward Time . . . . . . . . . . . . . . . . . . . . 65 3.3.3 Ergodic Quantum Processes from an Invertible Ergodic System . . . 66 3.3.4 Limiting Results for Ergodic Processes with Invertible Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 IID Case and Invariant Measures . . . . . . . . . . . . . . . . . . . . . . 70 CHAPTER 4 CONTINUOUS PARAMETER QUANTUM PROCESSES . . . . . . 76 4.1 Notations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Limiting Results in Forward Time . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Limiting Results in Backward Time . . . . . . . . . . . . . . . . . . . . . 83 4.4 Other Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 85 CHAPTER 5 LLN FOR ERGODIC QUANTUM PROCESSES . . . . . . . . . . . 89 5.1 Linear Cocyles and Lyapunov Exponent . . . . . . . . . . . . . . . . . . 89 5.2 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 vi 5.3 Limits for Fractional Contraction Coefficient . . . . . . . . . . . . . . . . 91 5.4 Proof of the Law of Large Numbers Theorem . . . . . . . . . . . . . . . 92 CHAPTER 6 CLT FOR ERGODIC QUANTUM PROCESSES . . . . . . . . . . . 99 6.1 Filtration and Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 Proof of the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . 101 6.3 Mixing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 vii CHAPTER 1 INTRODUCTION Quantum channels are used to describe the physical change of a quantum system over a unit interval of time. A quantum channel is a linear, completely positive (CP) and trace preserving (TP) map on the state space, i.e. a map on density matrices. Thus the evolution of the system via discrete time steps can be then obtained by iterative applications of a sequence of quantum channels. In finite dimension a quantum channel, φ, can be expressed in the Kraus form [48] X D φ(ρ) = Ki ρKi∗ i=1 with X D Ki∗ Ki = I . i=1 Where ( · )∗ denotes the adjoint operator. Therefore the evolution of the system at the n-th time step can be then obtained by X D ∗ ∗ ρ 7→ Kn,in . . . K1,i1 ρK1,i 1 . . . Kn,i n i1 ,i2 ,...in =1 where the dynamics at the j-th step is obtained by the quantum channel X D ∗ φj (ρ) = Kj,i ρKj,i . i=1 Therefore we may write the net change of an initial state ρ0 at the n-th time interval as ρn = φn ◦ . . . φ1 (ρo ) . As a consequence it is clear that the average change of an initial state ρ0 at any time step can be then described by the up to n compositions of the sequence of quantum channels (φi )i∈N defined as Φ (n) = φn ◦ . . . φi . 1 We shall call the sequence of up to n compositions, (Φ (n) )n∈N a quantum process and the original sequence of quantum channels (φi )i∈N the sequence describing the quantum process. In this thesis we study general classes of such quantum processes where the sequence describing the quantum processes has a strictly stationary distribution. 1.1 Random Compositions of Quantum Channels It is useful to know physical models where such random quantum channels and there- fore random quantum processes are used to study the time evolution of a physical system. 1.1.1 Random repeated interactions Suppose S is a small system with the Hilbert state space H ≡ CD and E the environ- ′ ment with Hilbert state space given by K ≡ CD . Let’s start with a single interaction of S with the environment E. Assuming that the initial state of the coupled system is in a product state σ = ρ ⊗ α where ρ is the initial state of the small system and α is the state of the environment we have that coupled system undergoes a unitary evolution U (following Schrödinger) σ 7→ U (ρ ⊗ α)U ∗ . Now if we are interested only in the state of the small system (as one often would) we take the partial trace to obtain the final state of S ρ 7→ TrK [U (ρ ⊗ α)U ∗ ]. If one is interested in repeated interactions of S with the environment E we obtain a sequence of quantum states of S after each interaction via ρn = TrK [Un (ρn−1 ⊗ αn )Un∗ where ρn−1 , ρn are the successive states of the system S, Un is the unitary transformation at the n-th step and αn the state of the auxiliary system E. In terms of quantum processes one can write ρn = Φ (n) (ρ0 ) 2 where ρ0 is the initial state and Φ (n) = φn ◦ . . . ◦ φ1 with φi (ρ) = TrK [Ui (ρ ⊗ αi )Ui∗ ]. We see that φi is indeed a quantum channel as it is written in its Stinespring representa- tion [66]. By certain assumptions on the distributions of Un and αn we see that φn and therefore Φ (n) can be considered as random processes. Such random repeated interactions are studied in [11, 57] for the IID case and [56, 55] for the ergodic case. 1.1.2 Repeated Measurements An important notion in theory of open quantum systems is indirect measurements. An indirect measurement on a quantum system is a measurement that is taken after some interaction with the system. When we make such a measurement the system changes by a unitary transformation (due to Schrödinger ), and then we update our knowledge of the system by conditioning on the measurement outcome (due to von Neumann). If we then repeat the indirect measurement indefinitely, we obtain a chain of random outcomes. Now consider the following scenario of repeated measurements where we do not record the measure outcome: At the (n − 1)-th time step; 1. We let the system evolve for some time, say tn . 2. An indirect measurement is performed and we do not record the measurement out- come (we “forget” the measurement). Then after the n-th time step the new state of the system is given by XD ρ 7→ Ki,n e−itn H ρeitn H Ki,n i=1 where ρ is the initial state and (Ki,n )D i=1 are the Karus operators of the generalized mea- surement at n-th time step. Similar to the repeated interactions we see that the dynamics of the system then can be described by compositions of quantum channels, i.e via a quan- tum process. 3 1.2 Positivity Preserving Maps: A More General Class Than Quantum Channels In the previous two examples we see that the quantum process was described by a sequence of quantum channels which are (in particular) trace preserving. However pos- itive maps that are not necessarily trace preserving are physically and mathematically interesting. 1. In a physical experiment one may be interested in an experiment where the system is discarded upon a certain measurement result. In this case the sequence describing the quantum process consists of completely positive trace non increasing maps φi , i.e. tr [φi (ρ)] ≤ tr [ρ] for any matrix ρ. Such linear super operators are historically called quantum operations. 2. Mathematically if one were to compute a moment generating function (MGF) for the sum of outcomes for a measurement process we do not have that the MGF is trace preserving. Actually, the MGF has no relation to the trace of the state: Let A be a finite set of measurement outcomes and let Ka be the (random) associated Karus operator for each outcome a ∈ A. Then after one measurement we have that the measurement outcome o1 is a with probability tr [Ka ρKa∗ ]. Now the quantum expectation of eto1 is given by X etλ(a) tr [Ka ρKa∗ ] a∈A where λ is map λ : A → R that embeds the set of measurement outcomes in R. Proceeding in this manner we see that for the sum of outcomes o1 , . . . on the quantum Pn expectation of et i=1 on is given by X Pn et i=1 λ(ai ) Kn,an . . . K1,a1 ρK1,a ∗ 1 ∗ . . . Kn,an = φnt ◦ . . . ◦ φ1t (ρ) a1 ...an ∈A where φjt (ρ) = tλ(a) K ρK ∗ . P a∈A e j,a j,a Here (Kj,a )a∈A is the set of (random) Kraus oper- ators at the j the measurement. Since the study of super operators that are not necessarily trace preserving is use- ful we opt to refer to any positivity preserving super operator as a quantum operation 4 therefore the term quantum operation will include maps that are not necessarily trace non-increasing. 1.3 Organization of the Chapters In chapter 1 we shall introduce some basic notations and theorems in probability the- ory, linear algebra, and quantum information theory that will be useful for the later chapters. In chapter 2 we generalize the results in [56, 55] to the stationary case and study some particularly special cases such as ergodicity and independently and identi- cally distributed (IID) cases. In chapter 3 we discuss the notion of a quantum process in continuous time parameter. Chapter 4 and 5 are dedicated to the study of quantum processes that are obtained by a sequence of ergodic quantum operations as discussed in [56, 55] and we obtain a law of large numbers and a central limit theorem for this case. Chapter 5 also contains conditions on mixing coefficient to guarantee the central limit theorem. 5 CHAPTER 2 PRELIMINARIES 2.1 Probability and Ergodicity Preliminaries A measurable space consists of a set, X, a σ -algebra of subsets of X, which we shall denote by X . A map f between two measurable spaces f : (X, X ) → (Y , Y ) is called measurable if f −1 (Y ) ⊆ Y . Whenever we want to specify the sigma algebras we shall call f a (Y , X )-measurable function. A measurable space equipped with a measure µ (a non-negative countably additive function defined on X ) shall be called a measure space. Therefore measure space is a triple (Ω, F , P) where Ω is the underlying set, F is a σ −algebra of subsets of Ω and P is the measure on Ω. A measure space (Ω, F , P) is called a probability space if P(X) = 1. When (Ω, F , P) is a probability space, the sets in F are called events and if the measure space is not necessarily a probability space we shall call sets in the σ -algebra as measurable sets. We refer the reader to [67] or [41] on basic results on measure spaces. For a measure space (X, X , µ) a property P is said to hold almost everywhere (a.e.) if there exists a set N ∈ X with µ(N ) = 0 and for all x ∈ X \ N we have the property P . In the context of a probability space we shall call P holds almost surely (a.s.) or that P holds with probability 1 (w.p. 1). A function f : (Ω, F , P) → (S, S) where (Ω, F , P) is a probability space and (S, S) a measurable space, is called almost surely measurable if there exists an event N with prob- ability 0 such that for all B ∈ S one has that f −1 (B) ∩ (Ω \ N ) ∈ F . We have that a function that is almost everywhere measurable is equal to a measurable function. Proposition 2.1.1. Let (Ω, F , P) be a probability space and let f be an almost surely measur- able function taking values in some measurable space (S, S). Then there exists a function f ′ that is measurable in the usual sense and f = f ′ almost surely. Proof. Let N be an event with probability 0 such that f −1 (B) ∩ (Ω \ N ) ∈ F for all B ∈ S. 6 Such a set must exist as f is almost surely measurable. Then pick any point s0 ∈ S define f ′ as follows    f (ω) if ω ∈ Ω \ N   ′  f (ω) =    s0 if ω ∈ N .   Then we have for any B ∈ S       N ⊔ (Ω \ N ) ∩ f −1 (B) if s0 ∈ B (f ′ )−1 (B) =     f −1 (B) ∩ (Ω \ N )  if s0 < B.   We have that f −1 (B) ∩ (Ω \ N ) ∈ F by the definition of f being almost surely measurable therefore we see that f ′ is measurable in the usual sense and it is clear that f = f ′ on N c which has probability 1. We note that some authors use the definition of almost sure measurable function in place of usual measurable function [17, p. 19]. It is useful to reduce the criterion of a function f : (X, X ) → (Y , Y ) to be measurable from f −1 (Y ) ⊆ X to a property on a gener- ating family on Y . To this end we present the following lemma. Lemma 2.1.2. Let f : (X, X ) → (Y , Y ) be a measurable function suppose A ⊆ 2Y with σ (A) = Y . Then f is (Y , X )-measurable if and only if f −1 (A) ⊆ X . Here 2Y denotes the set of all subsets of Y and σ (A) denotes the σ -algebra generated by A (i.e. intersection of all σ -algebras containing X). Proof. It is clear that f −1 (Y ) ⊆ X implies that f −1 (A) ⊆ X as σ (A) = Y implies that A ⊆ Y . For the opposite implication, first note that the class C of sets B ⊆ Y such that f −1 (B) ∈ X forms a σ -algebra of subsets of X: C = {B ⊆ Y : f −1 (B) ∈ X }. Now if f −1 (A) ∈ X for all A ∈ A we have that A ⊆ C but as C is a σ -algebra we have that σ (A) ⊆ C = {B ⊆ Y : f −1 (B) ∈ X }. Therefore f −1 (Y ) ⊆ X whence F is (Y , X )-measurable. 7 Remark 2.1.3. More generally we have that for a measurable function f : (X, X ) → (Y , Y ), a σ -algebra of subsets of X, L and G ⊆ 2X we have that f −1 (G) ⊆ L implies that f −1 (σ (G)) ⊆ L. Another useful Theorem is Monotone Class Theorem also known as the π − λ Theo- rem. It gives a criteria for two finite measures to be equal. First we need few definitions. 1. A π-system on Ω is a class of subsets of Ω which is closed under finite intersection. 2. A λ-system on Ω is a class of subsets of Ω which contains Ω and is closed under proper differences and increasing limits. We present the following version of the Monotone class theorem proof of which can be found in many standard probability theory textbooks such as [21]. Monotone Class Theorem 2.1.4. If C is a π-system then σ (C) = λ(C), where λ(C) denotes the smallest λ-system containing C. Remark 2.1.5. With this theorem we see that if two finite measures µ, ν on (Ω, F ) agree on a π-system, C, such that σ (C) = F then they must agree on F . This is because the class of sets L = {A ∈ F : µ(A) = ν(A)} forms a λ − system. Then the result follows from the monotone class theorem 2.1.4. A measurable map X from a probability space (Ω, F , P) to a measure space (S, S) in- duce a probability on the measurable space (S, S), denoted by P◦X −1 , henceforth referred to as the push-forward of P under X, defined by P ◦ X −1 (T ) = P[X −1 (T )] = P({ω ∈ Ω : X(ω) ∈ T }) ; ∀ T ∈ S. The push-forward is also known as the distribution or law of X. We may also denote the law of a random variable X simply as Law(X). We shall denote the integration of a measurable function f with respect to the proba- bility measure by E[f ]: Z f (ω) dP(ω) := E[f ]. Ω 8 For 1 ≤ p < ∞, we denote by Lp (Ω, F , P) (or simply Lp , when the underlying probability space is clear) to denote the equivalence class of measurable functions f : Ω → R such that E[|f |P ] < ∞ and for a function f ∈ Lp we define its Lp -norm by ∥f ∥p = (E[f p ])1/p . For p = ∞ we define L∞ to be the equivalence class of measurable functions f : Ω → R such that ess sup |f | < ∞ where ess sup f = inf{α ∈ R : P{ω : f (ω) > α = 0}} and we define the L∞ -norm of a function f as ∥f ∥∞ = ess sup |f |. Another important concept in probability theory is the notion of conditional expecta- tion. Below we list some results related to conditional expectation and direct the reader to any standard textbook on probability theory for a detailed treatment of conditional expectation. For ease we give some references here [41, 21]. Conditional Expectation 2.1.6. Let (Ω, F , P) be a probability space. Suppose A is a sub σ - algebra of F then for any f ∈ L1 (Ω, F , P) there exists g ∈ L1 (Ω, A, P) such that for any event A ∈ A one has E[1A f ] = E[1A g]. Furthermore g is a.s. unique. We shall denote g := E[f |A] and we also have that the map f 7→ E[f |A] is a bounded linear map from L1 (F ) → L1 (A) with ∥E[f |A]∥1 ≤ ∥f ∥1 . We also have the following properties: 1. if f ≥ 0 a.s then E[f |A] ≥ 0 a.s. 2. For 1 ≤ p ≤ ∞, if f ∈ Lp then we have thatE[f |A] ∈ Lp and ∥E[f |A]∥p ≤ ∥f ∥p . 9 3. If B is another sub σ -algebra of F with B ⊆ A then we have that E[f |B] = E[E[f |A]|B]. 4. If g ∈ L1 (A) then we have that E[f g|A] = gE[f |A]. 5. E[f E[g|A] = E[gE[f |A]]] = E[E[f |A] E[g|A]]. Given a probability space (Ω, F , P) a measurable map θ : (Ω, F ) → (Ω, F ) is called measure preserving if for all events E ∈ F one has the probability of the event E and the pre-image of E under the map θ have the same probability i.e. P[θ −1 (E)] = P(E) ; ∀E ∈ F . The probability space together with the measure preserving transformation, (Ω, F , P, θ) is called a measure preserving system. A map θ : (Ω, F ) → (Ω, F ) is called a bi-measurable au- tomorphism, if θ is invertible and if both θ and the inverse map θ −1 are measurable. Given a family of measure preserving transformations (θg )g∈G on a probability space (Ω, F , P) where G is some indexing set we give the following two definitions of ergodicity: Definition 2.1.7 (w-ergodicity). The family of measure preserving transformations (θg )g∈G is called weakly - ergodic if all essentially invariant measurable sets have trivial measures. That is for A ∈ F with P(θg−1 (A)∆A) = 0 for all g ∈ G implies that P(A) is 0 or 1. Definition 2.1.8 (s-ergodicity). The family of measure preserving transformations (θg )g∈G is called strongly - ergodic if all invariant measurable sets have trivial measures. That is for all measurable sets A with, θg−1 (A) = A for all g ∈ G implies that P(A) is 0 or 1. For a group (G, +) a family of measure preserving transformations indexed by G, {θg }g∈G , is called a one-parameter group of measure preserving transformations if we have 1. For all g1 , g2 ∈ G, θg1 ◦ θg2 = θg1 +g2 . 2. For the identity element e ∈ G, θe (ω) = ω for all ω ∈ Ω. With these two properties we must have that for all g ∈ G, ω 7→ θg (ω) is a bijection on Ω and that θg−1 = θ−g where −g is the inverse element of G under its binary operation + i.e. g + (−g) = e = (−g) + g. Hence the family {θg }g∈G consists of bi-measurable automorphisms on (Ω, F ) that are measure preserving. It is well known [33] that when G = N, these two 10 definitions are equivalent, without further assumptions on the probability space. In the case (Ω, F , P) is a standard Borel space (therefore F is the Borel sigma-algebra) and G is a separable and locally compact group by a theorem of Mackey [54] we have the two definitions are equivalent. For a case of a single measure preserving transformation θ we note that if θ is bi-measurable (i.e the inverse exists and also measurable) we must have that τ = θ −1 is also measure preserving since θ −1 (θ(E)) = E (as θ is injective) we have that P(E) = P(θ −1 (θ(E))) = P(θ(E)) for any E ∈ F where the last equality uses θ is measure preserving. Thus P(τ −1 (E)) = P(E). If θ is ergodic then so is τ = θ −1 . This is because for a θ −1 invariant subset A we have if θ(A) = A then by injectivity we have A = θ −1 (θ(A)) = θ −1 (A), that is A is invariant under θ −1 , too. We have the following consequence of ergodicity: Proposition 2.1.9. Given a probability space (Ω, F , P) with a a family of measure preserving transformations (θg )g∈G on Ω we have that a measurable function f : Ω → R ∪ {−∞} that is essentially θg -invariant for all g ∈ G (i.e. f ◦ θg = f almost surely for all g ∈ G) is almost surely constant. Proof. For a ∈ [−∞, ∞) let Xa = {ω : f (ω) ≥ a} then we have as f is essentially θg -invariant for all g ∈ G, Xa is also essentially θg -invariant for all g ∈ G. Thus P(Xa ) = {0, 1} for all a ∈ [−∞, ∞). Then we claim that f = c almost surely where c = supa∈[−∞,∞)] {P(Xa ) = 1}. To this end first note that the collection of sets Xa with P(Xa ) = 1 is non-empty as f takes values in [−∞, ∞). Now if c = −∞ we have that f = −∞ almost surely. For the case c , −∞ we have that for all n ∈ N, P(Xc−1/n ) = 1 thus f ≥ c − 1/n for all n whence f ≥ c. On the other hand we have for each n ∈ N that P(Xc+1/n ) = 0, by the definition of c. Thus f ≤ c almost surely. A simply alteration of the proof of above lemma gives us the following immediate corollary. 11 Corollary 2.1.10. Given a probability space (Ω, F , P) with a a family of measure preserving transformations (θg )g∈G on Ω we have that a measurable function f : Ω → R ∪ {∞} that is essentially θg -invariant for all g ∈ G is almost surely constant. 2.2 Stationary Sequences A sequence of random variables (Xn , n ≥ 0) is said to be a stationary sequence if for each k ∈ N the joint distribution of the sequence (Xn )n∈N is the same as the distribution of the shifted sequence (Xn+k )n∈N . Given a probability space (Ω, F , P) and a measure preserving transformation θ any measurable map X on Ω that takes values in R generates a stationary sequence Xn (ω) = X(θ n (ω)). We see that the sequence (Xn )n∈N is indeed stationary by observing that for E = {ω : (X1 (ω), . . . Xn (ω)) ∈ B)} one has P(E) = P(θ k−1 (E)) = P(Xk , . . . Xn+k−1 ∈ B) where B is a Borel set in Rn . In fact, given any stationary sequence that take values in a nice measurable space (S, S) one can construct a measure µ on the space (S N , S N ), where S N denotes the product σ −algebra (i.e. the σ -algebra generated by sets of the form E1 × E2 × . . . Ek × ∞ Q i=k+1 S, k ∈ N and Ei are events in S), such that the sequence Xn (ω) = ωn has the same distribution as Yn . Then for θ : (S N , S N ) → (S N , S N ) given by θ(ω1 , ω2 . . .) = (ω2 , ω3 . . .) is µ-measure preserving and that Xn (ω) = X(θ(ω)) for ω ∈ S N . The construction of the measure µ uses a method more commonly known as the Kolmogorov’s extension. More generally there are two methods of extensions of these measures to a product of infinite families of probability measures: Given an arbitrary collection of probability Q spaces (Ωt , Ft , Pt )t∈Γ and Ω = t∈Γ Ωt with F the σ -algebra on Ω generated by cylinder sets one may wish to construct a probability P on (Ω, F ) such that: 12 • The coordinate projections πt : Ω → Ωt are P-independent random variables and for each t ∈ Γ the push-forward of P ◦ πt−1 coincides with Pt or one may wish to construct P so that • For any subset β of Γ if we denote the partial product space by (Ωβ , Fβ ) and if we have that there is a probability measure Pβ on (Ωβ , Fβ ) we wish to construct P so that for any finite subset β of Γ we have that the push-forward measure P ◦ πβ1 co- incides with Pβ . Note that Pβ on the finite product space (Ωβ , Fβ ) is not necessarily the product probability (which is known to exist for product of finitely may proba- bility spaces). The latter case is more commonly referred to as the Kolmogorov’s extension and the for- mer simply as the product probability. The latter scenario is also referred to as the pro- jective limit of probability spaces (see [2, §35]), It is known that the former holds true for any arbitrary collection of probability spaces. We also list some references for the reader: For a general product of probability spaces we refer to [2, Theorem 9.1] or to [40] or for a countable product we refer to [13, §10.6] or [32, §38]. On the other hand the consis- tency condition in the second scenario is weaker than the independence requirement of the projections πt in the first scenario and as such require more structure on the prob- ability spaces. However if the probability spaces (Ωt , Ft , Pt )t∈Γ are sufficiently nice then the second scenario holds [46], [26, Chapter 45]. We list theorems addressing both cases below: Infinite Product Probability 2.2.1. Given a family of probability spaces {(Ωt , Ft , Pt )}t∈Γ there exists a probability measure P ˜ on the product space, i.e. the underlying set is Qt∈Γ Ωt and ⊗t∈γ Ft is the σ -algebra generated by finite unions of the sets of the form Bt1 × Bt2 . . . × Btn × Q t∈Γ \{t1 ,...,tn } Ωt with Bti ∈ Fti and n ∈ N, (such sets are called cylinder sets) such that for a set Q of the form Bt1 × Bt2 . . . × Btn × t∈Γ \{t1 ,...,tn } Ωt one has    Y  Y n P˜ Bt × Bt . . . × Bt × Ω t  = Pti (Bti ).  1 2 n   t∈Γ \{t1 ,...,tn } i=1 13 Daniell-Kolmogorov Extension Theorem 2.2.2. Let {(Ωt , Ft }t∈Γ be a family of measurable spaces. Assume in addition that each Ωt is equipped with a topology τt and assume that for each finite subset β ⊆ Γ there is a a probability measure Pβ that is inner regular with respect Q to the product topology on t∈β Ωβ . For subsets α ⊆ β of Γ let πβ→α denote the projection of Q Q t∈β Ωt into t∈α Ωt . If the probability measure Pβ satisfy the consistency condition −1 Pβ ◦ πβ→α = Pα for any two finite subsets α ⊆ β then there exists a unique probability measure P on the product Q  −1 space t∈Γ Ωt , ⊗t∈γ Ft such that for any finite subset β ⊆ Γ we have P ◦ πβ = Pβ . Remark 2.2.3. Some authors (see for an example [58, proposition V.1.1, proposition V.1.5]) credit above theorems to (a small generalization of) a theorem by Ionescu-Tulcea [38] (or to [50] for a restatement of the theorem, since the original is hard to find). We also note that we shall opt to call either of theses extended probability spaces (ΩΓ , F Γ , P) ˜ as the Kolmogorov’s extension, for ease. With above theorem we have the following useful results. Proposition 2.2.4. Let (Yn )n∈N be a stationary sequence on (Ω, F , P) taking values in a mea- surable space (S, S), then there exists a probability space (Ω′ , F ′ , P′ ) with a measure preserving d transformation θ and a measurable map X on Ω′ taking values in S such that X ◦ θ n = Yn , d where = denotes the equality in distribution. Proof. Using the law of each Yn we have that (S, S, P ◦ Yn−1 ) is a probability space. Then let (Ω′ , F ′ , P′ ) be the product probability space obtained via the Kolmogorov’s extension, i.e. (Πi∈N S, ⊗i∈N S, P). ˜ Then we define for each n ∈ N the following measurable map Xn : Πi∈N S → S Xn ((s1 , s2 , . . .)) = sn . Then we have that P ˜ ◦ Xn−1 = P ◦ Yn−1 . Now define the measure preserving transforma- tion θ : Ω′ → Ω′ as the shift operator θ((s1 , s2 . . .)) = (s2 , s3 . . .). To see that θ is indeed 14 measurable and measure preserving it is enough to establish that for sets of the form k ∞ i=1 Bi × i=k+1 S where each Bi ∈ S. This is because such sets generate the σ -algebra (see lemma 2.1.2) and are closed under finite intersections i.e. such sets form a π − system (see remark 2.1.5). For such sets we have that      k ∞  k ∞  −1  θ    Bi × S  = S ×   Bi × S i=1 i=k+1 i=1 i=k+1 and that k  ∞  Y k Y k   k  ∞  ˜ P Bi × S = P ◦ Y −1 (Bi ) = P ◦ Y −1 (Bi ) = P θ −1 Bi × S . i i+1 i=1 i=k+1 i=1 i=1 i=1 i=k+1 Here we have used that the sequence Yn is stationary in the second to last equality. Now d let X = X1 then we have that Xn = X1 ◦ θ n−1 = Yn . To get the exact results as stated in the proposition we may start with another copy of (S, S, P ◦ Y1−1 ) in the product space so that d d d X1 = Y1 , X2 = Y1 , X3 = Y2 , etc. It is quite useful to observe the following fact about stationary sequences Lemma 2.2.5. Any stationary sequence Xn , n ≥ 0 taking values in a measurable space (S, S) d can be extended in to a two-sided stationary sequence (Yn )n∈Z such that Yn = Xn for n ≥ 0. Proof. For each n ∈ Z we define a probability space (S, S, Pn ) by    −1 if n ≥ 0 P ◦ Xn    Pn =   −1  P ◦ X−n if n < 0   Then an obvious application of theorem 2.2.1 gives us the existence of a probability space ˜ such that (S Z , ⊗n∈Z A, P)     k  Y  Y  Y P˜  S  × A × . . . A × . . . A ×   −n 0 k  S  =  Pi (Ai )   i<−n i>k i=−n where each Ai ∈ S. Then consider the two sided sequence of measurable functions Yn : S Z → S such that Yn (s̄) = sn where s̄ = (. . . , s−1 , s0 , s1 . . .). Then we have that for n ≥ 0 and A∈S ˜ n ∈ A] = Pn (A) = P ◦ Xn−1 (A). P[Y 15 Therefore Yn and Xn have the same distribution for n ≥ 0. Furthermore (Yn )n∈Z has the same distribution as (Yn+k )n∈Z for any k ∈ N by the definition of Yn . Lemma 2.2.6. Let (Xn )n∈Z be a two-sided stationary sequence taking values in a measurable space (S, S) then there exists a probability space (Ω′ , F ′ , P), ˜ an invertible measure preserving transformation θ and a measurable map Y such that Y ◦ θ n = Xn for all n ∈ Z. Proof. Consider the collection of probability spaces given by (S, S, P ◦ Xn−1 )n∈Z and take the Kolmogorov’s extension of this family (S Z , ⊗z∈Z S, P). ˜ Then for any n ∈ Z define Yn ((. . . s1 , s0 , s2 , . . .)) = sn . d ˜ ◦ Yn−1 (B) = P ◦ X −1 (B) thus Yn = Xn . Now if Then for n ∈ Z, we have that for any B ∈ S, P N we define θ : S Z → S Z as θ(s̄)n = sn+1 for s̄ = (. . . , , s1 , s0 , s1 , . . .) we have that θ is measurable and measure preserving by verifying these properties on cylinder sets (similar to lemma 2.2.4). Furthermore we also have that θ is invertible (with a measurable inverse). We also d have that Yn = Yo ◦ θ n for any n ∈ Z and that Xn = Y0 ◦ θ n for any n ∈ Z. Lemma 2.2.7. Given a sequence of independently and identically distributed (I.I.D.) random variables (Xn )n∈N there exists a probability space (Ω, F , θ) with an ergodic measure preserving d transformation θ and a sequence of IID maps on Ω such that Xn = Yn and Yn = Y1 ◦ θ n . Q  Proof. Suppose Xn take values in (S, S) and consider the product space n∈N S, ⊗n∈N F , ˜ P d and let Yn (s1 , s2 . . .) = sn . Define θ(s1 , s2 , . . .) = s2 , s3 . . ., The proofs of the facts that Xn = Yn and Yn = θ n−1 ◦ Y1 are similar to previous results. To prove that θ is ergodic, let A be an invariant set, thus θ −1 (A) = A iterating we get θ −n (A) = A, for any n ∈ N thus A ∈ σ (Yn , Yn+1 , . . .). Therefore we have that A is in the tail σ -algebra. Then by Kol- mogorov’s zero-one law we have that P(A) is either 0 or 1. 2.3 Natural Extension of a Measure Preserving Transformation In this section we introduce a natural extension of a measure preserving transforma- tion that has reasonably nice properties so that the extended measure preserving trans- 16 formation is injective hence invertible. Suppose (Ω, F , P) is a probability space with a surjective measure preserving trans- formation θ then let Ω̃ be the space consisting of infinite sequences ω̄ = (ω0 , ω1 , ω2 , . . .) where ωi ∈ Ω and θ(ωi ) = ωi−1 for all i > 0. We note that this construction can be done as θ is assumed to be surjective. Now we transform Ω̃ in to a measurable space by intro- ducing a σ −algebra as follows: Let F˜ be the σ -algebra generated by sets of the form {ω̄ = (ω0 , ω1 , ω2 , . . .) ∈ Ω̃ : ωi ∈ E} where E ∈ F and i ≥ 0. Now we equip this measurable space with the following probability: For a set of the form Ẽ = {ω̄ = (ω0 , ω1 , ω2 , . . .) ∈ Ω̃ : ωi ∈ E} where E ∈ F and i ≥ 0 we define P( ˜ Ẽ) = P(E). Then from the definition of P ˜ we have that     ˜ {ω̄ : ω0 ∈ E0 , . . . ωk ∈ Ek } = P θ −k (E0 ) ∩ θ −k+1 (E0 ) . . . θ −1 (E0 ) ∩ E0 . P (2.3.1) Therefore we may apply Kolmogorov’s extension theorem to extend the definition of P ˜ to entirety of F˜ . Now on the probability space (Ω̃, F˜ , P) ˜ we define a transformation θ̃ as θ̃(ω̄) = (θ(ω0 ), θ(ω1 ), . . .). Then it is clear that θ̃ is invertible with the inverse defined as θ̃ −1 (ω̄) = (ω1 , ω2 , . . .). We verify that θ̃ and its inverse are measurable by considering pre-images on the generat- ing set of the σ -algebra F˜ (see lemma 2.1.2) and using that θ is measurable. Furthermore 2.3.1 shows that θ̃ is also a measure preserving transformation. We shall call this newly constructed probability space (Ω̃, F˜ , P) ˜ together with the measure preserving transfor- mation θ̃ the natural extension of (Ω, F , P, θ). We refer to [63, 16] for a full treatment of this construction. We have the following lemma form [16, p. 241]. Lemma 2.3.1. Let (Ω, F , P, θ) be a probability space endowed with a measure preserving trans- formation θ and let (Ω̃, F˜ , P, ˜ θ̃) be its natural extension as described above then θ is ergodic if and only if θ̃ is ergodic. 17 Remark 2.3.2. We note that the requirement that θ(Ω) = Ω in order for natural extension to exist can be relaxed in to θ(X) ∈ F . This is because if θ(X) is measurable then we can find the probability of the event θ(X) to be 1 as X ⊆ θ −1 (θ(X)) means that 1 = P(X) ≤ P(θ −1 (θ(X))) = P(θ(X)), where the last equality uses that θ is measure preserving. Thus we may restrict the probability in to the set θ(X) and to the sigma algebra F ∩ θ(X) in which case we have that the system can be inverted. Since the condition that θ(Ω) is an event is weaker than the assumption that θ(Ω) = Ω we now give some examples of such ergodic systems and notice that most nice systems satisfy this property. Lemma 2.3.3. Let X be a separable complete metric space (also known as a Polish space) and let B(X) be the σ −algebra of Borel sets of X then for a measurable function f : X → Rn (where we have equipped Rn with its Borel σ -algebra) we have that for any Borel subset B of X, f (B) is Lebesgue measurable. Proof. Consider the graph of the function f : G(f ) = {(x, y) ∈ X × Y : y = f (x)}. Firstly we claim the following known results in measure theory: 1. The product σ -algebra of two Polish spaces (X, B(X)) and (Y , B(Y )) (where B(X) and B(Y ) are the Borel σ -algebra generated by the respective metrics on X and Y ), i.e. B(X) ⊗ B(Y ) is the same as the Borel σ -algebra of the space X × Y [19, Proposition 4.1.7]: B(X) ⊗ B(Y ) = B(X × Y ). 2. For a measurable function f : (X, B(X)) → (Y , B(Y )) where (X, B(X)) is a topological space with its Borel σ -algebra and (Y , B(Y )) is a separable topological space with its Borel σ -algebra then we have that the graph of f is a measurable set in B(X × Y ) [49, pp. 384, 457]. Since Rn is a Polish space we have that the graph of f , G(f ) is a measurable set in B(X × Rn ) = B(X) ⊗ B(Rn ). Now consider the continuous function, π2 : X × Rn → Rn given by 18 π2 (x, y) = y, i.e. the projection on to the second coordinate. Then we have that for any Borel subset B ⊆ X π2 ({(x, y) : f (x) = y, x ∈ B}) = π2 (G(f ) ∩ (B × Y )) = f (B). This means that for any Borel set B in X, f (B) is a continuous image (i.e image under a continuous function) of a Borel subset and therefore we have that f (B) is an analytic set and therefore f (B) is a Lebesgue measurable set. We refer the reader to [42] for results on analytic sets or to [64] for results on Polish spaces and Borel sets. Equipped with lemma 2.3.3 we give an example of a class of an ergodic system that can be extended in to an invertible ergodic system. Example 2.3.4. Let θ be a measure-preserving transformation on [0, 1] equipped with the Lebesgue measure and the σ -algebra of Lebesgue measurable sets. Then θ([0, 1]) is measurable with full measure. This is because if θ is (L, L)-measurable then it is necessarily (L, B)-measurable, where we have opt to denote the Lebesgue σ -algebra by L and the Borel σ -algebra by B. Therefore we must have that θ is almost everywhere equal to a Borel (B, B)-measurable function. Therefore there exists a Borel set E with full Lebesgue measure and a Borel measurable function f such that f = θ on E. Moreover f (E) (and hence θ(E)) is analytic and hence Lebesgue measurable from lemma 2.3.3 and we have that E ⊆ θ −1 (θ(E)) and thus as θ is measure preserving we have that θ(E) has Lebesgue measure 1. This is because E ⊆ θ −1 (θ(E)) =⇒ 1 = λ(E) ≤ λ(θ −1 (θ(E))) = λ(θ(E)), where λ denote the Lebesgue measure. Then (θ([0, 1]))c is a null set and as Lebesgue measure is complete we have that θ([0, 1]) is a measurable set with full measure. We see from the remark 2.3.2 that this means that an ergodic system on [0, 1] can be embedded in an invertible ergodic system. 2.4 Matrices and Super Operators We denote by MD the space of D × D complex matrices. For any matrix A ∈ MD we denote by A∗ its adjoint operator, i.e. the conjugate transpose of the matrix A. Throughout 19 this thesis we shall equip the space MD with the Schatten 1-norm also know as the trace norm. It is useful to know several properties of Schatten p-norms. For A ∈ MD and p ≥ 1, we denote by ||A||p the Schatten p-norm of A. That is   p  1 ∥A∥p = Tr (A∗ A) 2 p . The Schatten ∞-norm is defined as: ∥A∥∞ = max{∥Au∥ : u ∈ CD , ∥u∥ ≤ 1}. Here the space of vectors CD is equipped with the usual vector 2−norm. Therefore the Schatten ∞-norm is the usual spectral norm of the operator A. We also note that the Schatten ∞-norm coincide with limp→∞ ||A||p , which justifies the use of the subscript ∞. We also note that the space MD is a Hilbert space with the Hilbert-Schmidt inner product defined as follows: For A, B ∈ MD ; ⟨A, B⟩ = tr[A∗ B]. We list some properties of the Schatten p-norm that will be useful later. 1. The Schatten p-norms are non-increasing in p, i.e. for any A ∈ MD and for 1 ≤ p ≤ q ≤ ∞, one has ∥A∥p ≥ ∥A∥q . (2.4.1) 2. For p ∈ [1, ∞] and for q such that 1 p + 1q = 1 (such q is called the conjugate of p) and for any A ∈ MD one has: |⟨A, B⟩| ≤ ∥A∥q ∥B∥p . (2.4.2) This is known as the Hölder’s inequality for Schatten norms. 3. For p = 2 one gets the Frobenius norm which is sometimes denoted by ∥ · ∥F . The case p = 1 is called the trace norm. Recall that for a positive semi-definite matrix trace norm and trace are equal. 20 4. For any three matrices A, B and C and for each p ∈ [1, ∞] ∥ABC∥p ≤ ∥A∥∞ ∥B∥p ∥C∥∞ . (2.4.3) In particular we have that for any p ∈ [1, ∞] Schatten p-norm is submultiplicative: ∥AB∥p ≤ ∥A∥p ∥B∥p . (2.4.4) 5. For every nonzero matrix A and for 1 ≤ p ≤ q ≤ ∞ we have that ∥A∥p ≤ rank A1/p−1/q ∥A∥q . (2.4.5) In particular we have that √ √ ∥A∥1 ≤ D∥A∥2 and ∥A∥2 ≤ D∥A∥∞ . (2.4.6) Proof of properties above can be found on most standard books on finite dimensional linear algebra. We list few references here [31, 3, 36]. Throughout this thesis we will use the Schatten 1-norm (i.e. the trace norm) to normalize the space MD and as such we will drop the subscript 1 and just write ∥ · ∥ to indicate the trace norm. Definition 2.4.1. Norm on MD will be Schatten 1-norm: h i ∥A∥ = tr (A∗ A)(1/2) ∀A ∈ MD . (2.4.7) It is convenient to introduce notation for certain subsets of positive semi-definite ma- trices as follows: 1. POSD is the set of all positive semi-definite D × D matrices, 2. POSoD is the set of all positive definite D × D matrices, 3. SD is the set of positive semi-definite D × D matrices with trace one, and 4. SoD is the set of positive definite D × D matrices with trace one. 21 The space of linear (super) operators on MD will be denoted by L(MD ). We equip the space L(MD ) with the operator norm induced by the trace norm on MD . That is for φ ∈ L(MD ): n o φ = sup φ(A) : A ∈ MD , ||A|| = 1 . (2.4.8) With this definition one can easily verify that for φ ∈ L(MD ) and A ∈ MD φ(A) ≤ φ ∥A∥. A map φ ∈ L(MD ) is called positive or positivity preserving if φ(POSD ) ⊆ POSD and strictly positive or positivity improving if φ(POSD \ {0}) ⊆ POSoD . A map φ in L(MD ) is called non- destructive if ker φ ∩ SD = ∅. A map φ in L(MD ) is called non-transient if ker φ∗ ∩ SD = ∅. For φ ∈ L(MD ) we define; v(φ) = inf{||φ(X)||Tr : X ∈ SD }. (2.4.9) Here we note that for any φ ∈ L(MD ) with the property that ker φ ∩ SD = ∅ (in particular for a strictly positive map) one must have that v(φ) > 0 because SD is a compact set under the topology induced by the trace norm and the map A 7→ φ(A) is a continuous function. For such φ we have that φ acts on SD and induces a transformation of SD , called the projective action of φ on SD , where the image of A ∈ SD under φ is denoted by φ · A and defined by φ(A) φ·A = . (2.4.10) φ(A) We also have that for φ and ψ with the property that both ker(φ) ∩ SD and ker(ψ) ∩ SD are empty (i.e. whenever the projective action of ψ and φ on SD are defined); (φ ◦ ψ) · A = φ · (ψ · A). (2.4.11) We may even extend the definition of the projective action to include any φ ∈ L(MD ) and any A ∈ MD be defining it as:    φ(A)    whenever φ(A) , 0  φ(A)  φ·A =  (2.4.12)   0 otherwise.   22 We define the following subsets of L(MD ) 1. P M denotes the set of all positive maps in L(MD ). 2. P M o denotes the set of all strictly positive maps in L(MD ). 3. K denotes super operator that are non-destructive. 4. K ∗ denote super operators that are non-transient. We see that the sets defined above are all Borel subsets on L(MD ) with respect to the topology generated by the operator norm in 2.4.8. Lemma 2.4.2. The sets P M, P M o , K and K ∗ are Borel sets in the space L(MD ) with the operator norm in 2.4.8. Proof. First we claim that P M is a closed set. To see this suppose (ϕn )n∈N is a sequence of positive maps that converges to ϕ in L(MD ). Then we have for any positive semi-definite matrix P , ϕ(P ) is positive semi-definite. Then for any positive semi-definite matrix Q we have that 0 ≤ ⟨Q, ϕn (P )⟩. Now taking the limits we have that ⟨Q, ϕn (P )⟩ → ⟨Q, ϕ(P )⟩ since ϕn converges to ϕ in norm. This establishes ϕ(P ) ∈ P OS o for any P ∈ P OS o thus ϕ is a positive map. Let ϕ be a strictly positive map in L(MD ). Then we have that there exists δ > 0 such that ⟨ϕ(P ), Q⟩ ≥ δ for all P , Q ∈ SD . Now for any positive map ψ with ψ − ϕ < δ/2 and for any P , Q ∈ SD , we have that |⟨(ψ −ϕ)P , Q⟩| ≤ ϕ − ψ . Thus ⟨ϕ(P ), Q⟩− ϕ − ψ ≤ ⟨ψ(P ), Q⟩. Hence δ/2 ≤ ⟨ψ(P ), Q⟩. Thus ψ is also strictly positive. This shows that P M 0 is relatively open in L(MD ). Now let ϕ ∈ K. Then as SD is a compact set in MD (under the trace norm) we have that there exists δ > 0 such that ∥ϕ(P )∥ ≥ δ for all P ∈ SD . Now let ψ ∈ L(MD ) such that ψ − ϕ < δ/2. Now for P ∈ SD we have that ψ(P ) − ∥ϕ(P )∥ ≤ ψ(P ) − ϕ(P ) < δ/2, Thus δ/2 ≤ ∥ϕ(P )∥ − δ/2 ≤ ψ(P ) . Hence P < ker ψ. This establishes that K is an open set. The proof of K ∗ is open is similar. 23 2.5 Quantum Channels and Quantum Operations A quantum channel is a map φ ∈ L(MD ) such that φ is trace preserving and completely positive (TPCP). A map φ ∈ L(MD ) is called completely positive if for all n ∈ N the map φ ⊗ id : MD ⊗ Mn → MD ⊗ Mn is a positive map. It is natural to identify MD ⊗ Mn as the C ∗ -algebra of n × n matrices with entries from MD . Then φ ⊗ id acts as      11 . . . A1n  φ(A11 ) . . . φ(A1n ) A        .. . .   . .. .   .  . . .  7→  .. . . . .         An1 . . . Ann  φ(An1 ) . . . φ(A )  nn where each Aij ∈ MD . It is well known [59, 68, 69] that any quantum channel admits a representation known as Kraus form of the quantum channel. We mention here that the Kraus representation is not unique. Kraus representation 2.5.1. Any linear map φ ∈ L(MD ) is completely positive if and only if it admits a representation of the form X n φ(X) = Ki XKi∗ (2.5.1) i=1 and the map is trace preserving if and only if X n Ki∗ Ki = I. i=1 In quantum measurements a positive-operator-valued measure (POVM) is a measure that takes values in the set of positive semi-definite matrices. POVMs are the most gen- eral kind of measurements in quantum theory. A POVM with finite number of positive operators, say {Pi }ni=1 must necessarily satisfy X n Pi = I. i=1 The probability of obtaining outcome i for a state ρ is given by prob(i|ρ) = tr [Pi ρ] . 24 However, we are unable to specify the post-measurement state after a POVM measure- ment. In order to do that we need more information on the measurement operators. For this case we need to decompose Pi as Pi = Ki∗ Ki . With this description we are able to spec- ify the state-change processes: If after the POVM measurement the outcome i is obtained then the initial state ρ changes to Ki ρKi∗ ρ 7→ h i. tr (Ki ρKi∗ ) Generally, a quantum operation is known as a completely positive map that is trace non-increasing: tr [(φ(ρ))] ≤ tr [(ρ)]. One may think of such a map as a measurement process where the quantum system is discarded after a certain measurement outcome. We also note that completely positive maps that are not trace preserving are mathematically interesting: Suppose one is to compute the moment generating function (MGF) is some observable after one (or possibly a sequence) of measurements. Then we have that for α∈R r (j) (j) (j)∗ X φα (ρ) = eαλi Ki ρKi . i=1 Here λi denotes the outcomes (embedded in R), r is the number of outcomes and we have that φα is a completely positive map that has no relation to the trace of ρ. For these reasons we opt to refer to completely positive map (or even just positive map) that are not necessarily trace preserving as quantum operation thought this thesis. We also note that such maps are of some interest in certain subfields of quantum theory: [65, 7, 1, 70, 24, 25]. 2.6 Geometry on States In this section we define a metric on the set of quantum states, SD . These results are primarily from [56, 55] and are similar to the Hennion metric in [34] as such we will refer to this metric as the Hennion metric. For any X, Y ∈ SD define m(X, Y ) = sup{λ : λY ≤ X}. (2.6.1) 25 Then the Hennion metric on SD , denoted by d ( · , · ), is defined as 1 − m(X, Y )m(X, Y ) d (X, Y ) = ; ∀ X, Y ∈ SD . (2.6.2) 1 + m(X, Y )m(X, Y ) Here we list some of the useful results related to the quantity m and the metric d. Proposition 2.6.1. For A, B, C ∈ SD we have 1. m(A, B) ∈ [0, 1]. 2. m(A, C)m(C, B) ≤ m(A, B). 3. m(A, B)m(B, A) = 1 if and only if A = B. 4. m(A, B) > 0 if A ∈ SoD . 5. If B ∈ SoD and An is a sequence in SD that converges to A in trace norm, then we have that lim m(An , B) = m(A, B) and lim m(B, An ) = m(B, A). n→∞ n→∞ ( ) tr [XA] 6. m(A, B) = min : X ∈ SD and tr [AY ] , 0 and also ( tr [XB] ) tr [XA] o m(A, B) = inf : X ∈ SD . tr [XB] Proof. It is clear that m(A, B) ≥ 0 for any two states A and B as states by definition are positive semi-definite matrices. To obtain the upper bound on m notice that if λB ≤ A then for any X ∈ SD we have that λtr [XB] ≤ tr [XA]. Therefore we have that ( ) tr [XA] m(A, B) ≤ inf : X ∈ SD and tr [XB] , 0 . tr [XB] 1 tr[IA] Now as D I ∈ SD and tr[IB] = 1 we have that m(A, B) ≤ 1. For part 2, if λC ≤ A and λ′ B ≤ C we must have that m(A, B) ≥ λλ′ and therefore we have that m(A, C)m(C, B) ≤ m(A, B). For part 3, in order for m(A, B)m(B, A) = 1 we must have that m(A, B) = 1 and m(A, B) = 1 due to part 1. And thus X = Y . Now if A ∈ SoD we have that A ≥ δI for some δ > 0 and as B is positive semi-definite with trace 1 we must have that λB ≤ A for some small positive λ yielding part 4. 26 For part 5 since B ∈ SoD we have that B ≥ δI for some δ > 0. Let ϵ be arbitrary then as limn→∞ ∥An − A∥ = 0 we have that there exits N ∈ N such that ∀n ≥ N , ∥An − A∥ ≤ ϵ. Let λ ≤ m(A, B), then we have that λB ≤ A, therefore for all n ≥ N we have that λB ≤ A ≤ An + ϵI ≤ An + (ϵ/δ)B. Therefore for n ≥ N we have that (λ − (ϵ/δ))B ≤ An . This gives us that ϵ lim inf m(An , B) ≥ m(A, B) − . n δ Conversely if λ ≤ lim supn m(An , B), then we have that λB ≤ Ank ≤ A + ϵI along a sub- sequence nk → ∞. Therefore we have that (λ − (ϵ/δ))B ≤ A which gives us ϵ lim sup m(An , B) ≤ m(A, B) + . n δ Since ϵ is arbitrary we have, by combining above two inequalities, that limn→∞ m(An , B) = δ ϵ m(A, B). Let t = ϵ+δ then t ∈ (0, 1), 1 − t = ϵ+δ and for λ ≤ m(B, A), we have that tλAn ≤ tλ(A + ϵI) ≤ tB + tϵI = tB + (1 − t)δI ≤ B for all n ≥ N . Thus lim inf m(B, An ) ≥ tm(B, A). n Conversely if λ ≤ lim supn m(B, An ), we have that tλA ≤ tλ(Ank + ϵI) ≤ B along some sub- sequence nk → ∞. Hence, t · lim sup m(B, An ) ≤ m(B, A). n Combining the two inequalities we see that limn→∞ m(B, An ) = m(B, A). Finally for part 6 we immediately see that if λB ≤ A then λtr [XB] ≤ λtr [XA] for any tr[XA] X ∈ SD now if tr [XB] , 0 we have that λ ≤ tr[XB] . Thus ( ) tr [XA] m(A, B) ≤ min : X ∈ SD and tr [XB] , 0 . tr [XB] Now we show that the infimum is attained. If λ = m(A, B) then we must have that A − λB must have 0 as an eigenvalue with a vector V ⃗ with BV⃗ , 0. Let X be the matrix v⃗ v⃗ , then we have that tr [XB] , 0 and tr [XA] = λtr [XB], with tr [XB] , 0. 27 Now if the infimum is restricted to the subset SoD we clearly have that ( tr [XA] m(A, B) ≤ inf : X ∈ SoD } (2.6.3) tr [XB] Now let λ > m(A, B) then we have that λ > 0 and that λB ≰ A. Since λB ≤ A if and only if for all vectors v⃗ we have that ⟨⃗ v , (λB − A)⃗ v ⟩ is non-positive real number, we must have that there exists a vector v⃗ such that ⟨⃗ v , (λB − A)⃗ v ⟩ > 0, i.e. λ⟨⃗ v , B⃗ v , v⃗A⟩. But since A v ⟩ > ⟨⃗ is positive semi-definite and as λ , 0 we must have that ⟨⃗ v ⟩ is positive (> 0) (and as A v , B⃗ is positive semi-definite ⟨⃗ v ⟩ ≥ 0). Now let M = v⃗ v⃗ + δI where δ is small so that M v , A⃗ is (strictly) positive definite. Now we consider cases: λ = 1, λ > 1 and λ < 1. If λ = 1 we have that ⟨⃗ v , B⃗ v , v⃗A⟩ and therefore we have ⟨⃗ v ⟩ > ⟨⃗ v ⟩ + δ > ⟨⃗ v , B⃗ v , v⃗A⟩ + δ > 0. So tr [MA] ⟨⃗ v , A⃗ v⟩ + δ = < 1 = λ. tr [MB] ⟨⃗ v , B⃗ v⟩ + δ Now if λ > 1 we have that λδ + λ⟨⃗ v , B⃗ v ⟩ > λ⟨⃗ v ⟩ + δ > ⟨⃗ v , B⃗ v , A⃗v ⟩ + δ > 0 and therefore tr [MA] ⟨⃗ v , A⃗v⟩ + δ = < λ. tr [MB] ⟨⃗ v⟩ + δ v , B⃗ λ⟨⃗ v ,B⃗v ⟩−⟨⃗ v ,⃗ v A⟩ For λ < 1 (but we still have λ > 0) we make δ so small so that 0 < δ < 1−λ then for such δ we have that ⟨⃗ v ⟩ + δ < λ⟨⃗ v , A⃗ v , B⃗v ⟩ + λδ. Therefore we have that tr [MA] ⟨⃗ v , A⃗v⟩ + δ = < λ. tr [MB] ⟨⃗ v⟩ + δ v , B⃗ Considering all cases we have found that whenever λ > m(A, B) then it must be the case that ( ) tr [XA] o λ > inf : X ∈ Sd . tr [XB] ( ) tr [XA] Therefore whenever λ ≤ inf : X ∈ Sod it must be the case that λ ≤ m(A, B). This tr [XB] could only happen if ( tr [XA] m(A, B) ≥ inf : X ∈ Sod } (2.6.4) tr [XB] Combining 2.6.3 and 2.6.4 we get the desired property. Proposition 2.6.2. Let A, B ∈ SD then we have that 28 1. d ( · , · ) is a metric on SD . t+ −t− 2. d (A, B) = t− +t+ −2t− t+ whenever A , B and where t+ and t− are largest and smallest real numbers such that tA + (1 − t)B ∈ SD resp. 3. d (A, B) ≥ 21 ∥A − B∥. 4. supA,B∈SD {d (A, B)} = 1. 5. If A ∈ SoD and B ∈ SD then d (A, B) = 1 if and only if B < SoD . 6. And if d1 (A, B) = ∥A − B∥ denotes the trace norm metric on SD we have that the spaces (SoD , d) and (SoD , d1 ) are homeomorphic. Proof. The symmetry and positivity of d is clear from proposition 2.6.1. It is also clear that d(A, A) = 0 by proposition 2.6.1. Now to prove the triangle inequality observe that the function f (x) = (1 − x)/(1 + x) we have that for x, y ∈ [0, 1] 2(1 − xy) 1 − xy 2 1 − xy f (x) + f (y) = = · x+y ≥ = f (xy). 1 + x + y + xy 1 + xy 1 + 1+xy 1 + xy 2 Where the inequality above uses that x+y ≥ 1 whenever x, y ∈ [0, 1]. This combined 1+ 1+xy with part 2 of proposition 2.6.1 gives us the triangle inequality, concluding the proof of part 1. Let t+ = sup{t : tA + (1 − t)B ∈ SD } and t− = inf{t : tA + (1 − t)B ∈ SD }. Since A, B ∈ SD we must have that t− ≤ 0 ≤ 1 ≤ t+ . Suppose t+ = 1 and assume that m(A, B) = λ > 0. Then we have that A−λB is in SD and thus for t = 1+λ we have that tA+(1−t)B = A−λB+λA ∈ SD which contradicts the maximality of t+ hence m(A, B) = 0 whence d (A, B) = 1. Similarly we have that if t− = 0 then d (A, B) = 1. Now for the case t+ > 1 and t− < 0 let A± = t± A + (1 − t± )B then we must have that A± ∈ SD \ SoD . Because, say for an example, if A+ ∈ SoD then A+ + δ(A − B) ≥ 0 for some small enough positive δ, which contradicts the maximality of t+ as 0 ≤ A+ + δ(X, Y ) = (t+ + δ)A + (1 − (t+ + δ))B. Hence both A− and A+ must have non trivial kernels. Furthermore we have that ker A+ ⊈ ker A− and vise versa. This because, if say for an example that ker A+ ⊆ ker A− then we can find a small positive δ such that A+ − δA− ≥ 0. Which gives a contradiction to the maximality of t+ 29 as 0 ≤ A+ − δA− = (t+ − δt− )A + (1 − (t+ − δt− ))B − δB ≤ (t+ − δt− )A + (1 − (t+ − δt− ))B. This concludes the proof of part 2. To prove part 3, notice that from the proof of part 2 we have that A+ t+ (t− − 1) + A− t+ (1 − t+ ) A= t+ (t− − t+ ) and A+ t− − A− t+ B= . t− − t+ Thus, A− + A+ A−B = . t+ − t− 1 Which gives us that ∥A − B∥ ≤ 2 . But from part 2 we have that t+ − t− t+ − t− 1 d (A, B) = ≥ t− + t+ − 2t− t+ t+ − t− where we have used that (t+ − t− )2 ≥ t− + t+ − 2t− t+ . Therefore we have that 1 d (A, B) ≥ ∥A − B∥ 2 finishing the proof of part 3. It is clear from the definition of m and d that 0 ≤ d (A, B) ≤ 1. To prove that the diameter of SD under the metric d is 1 we find A, B ∈ SD such that m(A, B) = 0 for an 1 example we may take A = DI and any B ∈ SD \ S0D , or more generally A ∈ SoD and any B ∈ SD \ S0D , which also proves part 4. We prove part 5 by showing that for B ∈ SoD and A ∈ SD with a sequence An in SD that converges to A in trace norm we must have the convergence in the Hennion metric and vise-versa. The first part of the claim follows from the convergence of m in part 5 of proposition 2.6.1 and the second part comes from part 3 of this lemma as ∥A − B∥ ≤ d (A, B). 30 Corollary 2.6.3. From the proof of proposition 2.6.2 we see that For For X, Y ∈ SD if X = a1 A+ + a2 A− and Y = b1 A+ + b2 A− where A± is defined as in the proof of proposition 2.6.2 above, then we must have that |a1 b2 − a2 b1 | d (X, Y ) = . a2 b1 + a1 b2 Proof. If X = a1 A+ +a2 A− then we have that a1 +a2 = 1 because X ∈ SD and by the definition of A+ and A− we must have that X is inside the line segment connecting A− with A+ . Similarly b1 + b2 = 1 and therefore we have that X − A− Y − A− a1 = and b1 = . A+ − A− A+ − A− Now using that A± = t± X + (1 − t± )Y we have that 1 − t− t+ − 1 −t− t+ a1 = , a2 = , b1 = and b2 = . t+ − t− t+ − t− t+ − t− t+ − t− Now replacing t± we have that |a2 b1 − a1 b2 | d (X, Y ) = . a2 b1 + a1 b2 2.7 Positive Super Operators Proposition 2.7.1. A map φ ∈ L(MD ) is positive (resp. strictly positive) if and only if the map φ∗ is positive (resp. strictly positive). Proof. Let X and Y be positive semi-definite matrices in MD . Then as φ is a positive map we have 0 ≤ ⟨Y , φ(X)⟩ = ⟨φ∗ (Y ), X⟩. (2.7.1) Since above is true for all X, Y ∈ POSD we must have that φ∗ (X) is positive semi-definite for all X ∈ POSD . Thus φ∗ is also a positive map. Similarly for the strictly positive case one has that the above inequality changes to 0 < ⟨Y , φ∗ (X)⟩ whenever X and Y are nonzero positive semi-definite matrices. This yields the desired result. 31 Proposition 2.7.2. Suppose the positive map φ has the property that ker φ∗ ∩ SD = ∅. Then we have that φ(POSoD ) ⊆ POSoD . Proof. Since for a matrix X we have that X ∈ POSoD if and only if there exists δ > 0 such that X ≥ δI we have that φ(POSoD ) ⊆ POSoD if and only if φ(I) is strictly positive definite. Thus assume toward a contradiction that φ(I) ∈ POSD \POSoD . Then φ(I) has a non trivial kernel. Let P be the projection on to the kernel of φ(I), Then we have that 0 = Tr(P φ(I)) = ⟨P ∗ , φ(I)⟩ = ⟨P , φ(I)⟩ = Tr(φ∗ (P )) = φ∗ (P ) . (2.7.2) This contradicts the assumption on the kernel of φ∗ Proposition 2.7.3. For a positive map φ, ker φ∗ ∩ SD = ∅ if and only if there exists Z ∈ POSD such that φ(Z) ∈ POS0D . Proof. Suppose φ has the given property on its kernel. Then by lemma 2.7.2 we have that φ(X) is a strictly positive definite matrix for any X which is also strictly positive. Conversely assume that there is Z ∈ POSD such that φ(Z) ∈ POSoD . Then assume that ker φ∗ contains a positive semi-definite matrix X of of trace 1. Thus, 0 = ⟨φ∗ (X), Z⟩ = ⟨X, φ(Z)⟩ > 0. (2.7.3) Which is a contradiction. We now present a lemma that will be useful in later chapters. Recall the definition of v(φ) for a map φ ∈ L(MD ) from (2.4.9). Lemma 2.7.4. Let φ ∈ L(MD ) be a positive map with the property that ker φ ∩ SD = ∅. Then for all X, Y ∈ SD ; φ ln φ(X) − ln φ(Y ) ≤ 2 d (X, Y ) . (2.7.4) v(φ) Proof. Let g : (SD , ∥·∥) → R be defined as g(X) = φ(X) . Since φ is positive with no matrix in SD in its kernel we must have that g(X) > 0 for all X ∈ SD . Since SD is closed in the 32 topology induced by the (trace) norm. We have that v(φ) = min{ φ(Z) : Z ∈ SD > 0}. (2.7.5) Now consider the map ln : (0, ∞) → R. We then have by mean value inequality that; φ(X) − φ(Y ) φ ∥X − Y ∥ ln φ(X) − ln φ(Y ) ≤ ≤ . (2.7.6) v(φ) v(φ) The result follows from proposition 2.6.2 as ∥X − Y ∥ ≤ 2d (X, Y ). Proposition 2.7.5. A map φ ∈ L(MD ) is strictly positive if and only if there exist t ∈ (0, 1] and a positive map ψ ∈ L(MD ) such that φ = (1 − t)ψ + tΩ. (2.7.7) Here Ω is the completely depolarizing quantum channel given by I Ω(A) = Tr(A) (2.7.8) D Proof. Assume that φ = Ω then we can easily see that Ω is strictly positive since one has that for nonzero positive semi-definite matrix A, tr(A) > 0. This proves the case for t = 1. Now if φ = (1 − t)ψ + tΩ for some t ∈ (0, 1) then we have that for any nonzero positive semi-definite matrix A, (1 − t)ψ(A) is positive semi-definite and Ω(A) is positive definite. Since the sum of a positive definite matrix and a positive semi-definite matrix is positive definite we have the desired result. Therefore if φ = (1 − t)ψ + tΩ for some t ∈ (0, 1] and ψ a positive map, we must have that φ is strictly positive. Conversely, assume that φ is strictly positive. Then we must have that for each nonzero rank one projection, P , the map P 7→ min σ (φ(P )) is continuous and takes values on (0, ∞). Here σ denotes the spectrum. Now as the set of rank1 1 projections is a compact set we have that there exists a > 0 such that a = min{min(σ (φ(P ))) : P is a rank 1 projection.}. Therefore we have that for X ∈ SD , there exists a rank 1 projection P such that X ≥ ||X||∞ P . Since φ(P ) ≥ min(σ (φ(P )))I we have that φ(X) ≥ ∥X∥∞ φ(P ) ≥ ∥X∥∞ aI ≥ a∥X∥I. (2.7.9) 33 Therefore on SD we have that for X ∈ SD , φ(X) ≥ aI. Thus we can find t sufficiently small such that t ∈ (0, 1) and t ≤ aD. Then define ψ ∈ L(MD ) by φ − tΩ ψ= . (2.7.10) 1−t We claim that ψ is a positive map whence the desired result follows. Let X be a positive semi-definite matrix. Then we have that 1 φ(X) − tΩ(X) = φ(X) − tTr(X)I ≥ φ(X) − aTr(X)I ≥ 0. (2.7.11) D Hence ψ is a positive map. Notice that the last inequality uses that for a positive semi- definite matrix trace and trace norm are equal. Above result gives a method to construct strictly positive maps through known posi- tive maps. Another method to construct strictly positive maps uses the Choi representa- tion of maps on L(MD ). More generally if we denote by T (n, m) the space of linear maps φ : L(Mn ) → L(Mm ) the Choi representation provides a one-to-one correspondence be- tween T (m, n) and Mm ⊗Mn  Mmn . For any choice of m, n ∈ N one may define a mapping J : T (n, m) → Mm ⊗ Mn  Mmn as X J(φ) = φ(Ei,j ) ⊗ Ei,j . (2.7.12) i,j∈{1,2...n} Here Ei,j denotes the matrix with entry 1 at row i and column j and all other entries are 0. One can prove that J is a linear bijection by noticing that given a matrix in Mmn denoted by J(φ) one can recover the mapping φ by the means of the equation φ(X) = 1L(Mm ) ⊗ Tr J(φ)(Im ⊗ X T ) .   (2.7.13) Here X T denotes the transpose matrix. We denote the inverse of this bijection by J −1 . Our next result states that for any positive definite matrix D ∈ Mmn , J −1 (D) is a strictly positive map. It is also well known that φ is completely positive if and only if J(φ) is positive semi-definite [12, 39]. These results can also be found on standard text books on Quantum Information Theory [59, 68]. 34 Lemma 2.7.6. For positive-definite D ∈ Mmn , J −1 (D) is a strictly positive map. Proof. Let D be a positive definite matrix in Mmn . Then there exists ϵ > 0 such that D − ϵI ≥ 0 (that is, D − ϵI is positive semi-definite). Then we have that the corresponding linear map, J −1 (D − ϵI) is completely positive by Choi-Jamilkowski theorem [12, 39]. In particular J −1 (D − ϵI) is a positive map. Now, since J and hence J −1 is linear we have J −1 (φ) = J −1 (D − ϵImn ) + ϵJ −1 (Imn ). (2.7.14) Let ψ = J −1 (I). Then we have by 2.7.13, for any X ∈ MD ψ(X) = 1L(Mm ) ⊗ Tr Imn (Im ⊗ X T ) = Im tr(X).   (2.7.15) Thus, ψ is clearly a strictly positive map. This tells us that J −1 (D) is the sum of a positive map and a strictly positive map. Therefore J −1 (D) is a strictly positive map. 2.8 Contraction Coefficient For any map φ ∈ L(MD ) we define the quantity c (φ), called the contraction coefficient of φ as follows c (φ) = sup{d (φ · A, φ · B) : A, B ∈ SD }. (2.8.1) We note that we are using the notation of projective action in the broader sense as defined in 2.4.12. We have the following properties of the contraction coefficient [55, Lemma 3.14] Lemma 2.8.1. Let φ ∈ L(MD ) be a positive map such that ker(φ) ∩ SD = ∅. Let X, Y ∈ SD Then, 1. d (φ · X, φ · Y ) ≤ c (φ) d (X, Y ). 2. c (φ) ≤ 1 and if φ is strictly positive then c (φ) < 1. 3. If there exist X, Y such that φ · X is (strictly) positive definite and φ · Y ∈ POSD \ POSoD then c (φ) = 1. 4. For a positive map ψ with ker(ψ) ∩ SD = ∅ we have c (φ ◦ ψ) ≤ c (φ) c (ψ). 35 5. If φ∗ also has the property that ker(φ∗ ) ∩ SD = ∅ then c (φ) = c (φ∗ ). Proof. If φ · X = φ · Y then we have that 0 = d (φ · X, φ · Y ) ≤ c (φ) d (X, Y ) trivially. Now assume that φ · X , φ · Y . Similar to the proof of proposition 2.6.2 we define the t+ and t− to be the largest and smallest values of t such that tX + (1 − t)Y ∈ SD resp. and s+ and s− to be the largest and smallest values of s such that sφ · X + (1 − s)φ · Y ∈ SD resp. Now let A± = t± X + (1 − t± )Y and B± = s± φ · X + (1 − s± )φ · Y . Then we have that φ(A+ ) φ · A+ = φ(A+ ) t+ φ(X) φ · X + (1 − t+ ) φ(Y ) φ · Y = φ(A+ )   t+ φ(X)  (1 − t+ ) φ(Y )  = φ · X +   φ · Y . φ(A+ ) φ(A+ )  Now as φ is a positive map we have that φ(A+ ) is a positive semi-definite matrix and so are φ(X) and φ(Y ) therefore we have that φ(A+ ) = t+ tr [(φ(X))] + (1 − t+ )tr [(φ(Y ))] = t+ φ(X) + (1 − t+ ) φ(Y ) . This gives us that   t+ φ(X)  t+ φ(X)  φ · A+ = φ · X + 1 −  φ · Y . (2.8.2) φ(A+ ) φ(A+ )  t+ ∥φ(X)∥ t+ ∥φ(X)∥ Now by definition of s± we must have that s+ ≥ ≥ s− . Let l1 = then direct ∥φ(A+ )∥ ∥φ(A+ )∥ computation yields that ! ! l1 − s − s + − l1 φ · A+ = B + B . s+ − s− + s+ − s− − Similarly we have that   t− φ(X)  t− φ(X)  φ · A− = φ · X + 1 −  φ · Y , (2.8.3) φ(A− ) φ(A− )  t− ∥φ(X)∥ t− ∥φ(X)∥ with s+ ≥ ≥ s− and for l2 = we have that ∥φ(A− )∥ ∥φ(A− )∥ ! ! l2 − s − s + − l2 φ · A− = B + B . s+ − s− + s+ − s− − 36 This means that φ maps the span of {A+ , A− } (inside SD ) in to the span of {B+ , B− } (in- side SD ) and that the matrix representation of the map w.r.t to the domain basis {A+ , A− } and range basis {B+ , B− } has all non-negative elements:   1 t+ φ(X) − s− φ(A+ ) t− φ(X) − s− φ(A− )     . s+ − s− s φ(A ) − t φ(X) s+ φ(A− ) − t− φ(X)   + + + Denote the above matrix by   m  11 m12   M =   .  m 21 m22 We also note that m11 m22 + m21 m12 > 0. This is because mij ≥ 0 so if m11 m22 + m21 m12 = 0 then either a row or a column of the matrix M must be 0. Now if M has a 0 column then one of φ(A± ) must be 0 (because s− ≤ 0 < 1 ≤ s+ we have s+ , s− ) which contradicts that ker(φ) ∩ SD = ∅. Now if M has a 0 row then φ(A± ) are proportional to one of B± . This can be seen, for an example, if t+ φ(X) − s− φ(A+ ) = 0 and t− φ(X) − s− φ(A− ) = 0 then we have from 2.8.2 and 2.8.3 that φ · A+ = s− φ · X + [1 − s− ] φ · Y and φ · A− = s− φ · X + [1 − s− ] φ · Y . WLOG assume that φ(A± ) are proportional to B+ then we have that φ(A± ) lies in the line connecting B− and B+ and also on the line connecting B+ with 0. But these lines only intersect at B+ therefore we have that both φ(A± ) are equal to B+ meaning that φ(X) = φ(Y ), another contradiction. Now let X = a1 A+ + a2 A− and Y = b1 A+ + b2 A− then from corollary 2.6.3 we have that φ(X) = a1 φ(A+ ) + a2 φ(A− ) = a1 (m11 B+ + m21 B− ) + a2 (m12 B+ + m22 B− ) and φ(Y ) = b1 (m11 B+ + m21 B− ) + b2 (m12 B+ + m22 B− ). 37 Therefore from corollary 2.6.3 we have that |(a1 m21 + a2 m22 )(b1 m11 + b2 m12 ) − (a1 m11 + a2 m12 )(b1 m21 + b2 m22 )| d (φ · X, φ · Y ) = (a1 m21 + a2 m22 )(b1 m11 + b2 m12 ) + (a1 m11 + a2 m12 )(b1 m21 + b2 m22 ) where the normalization by φ(X) and φ(Y ) have canceled out. Then rearranging we see that |(a2 b1 − a1 b2 )(m11 m22 − m21 m12 )| d (φ · X, φ · Y ) = (a2 b1 + a1 b2 )(m11 m22 + m21 m12 ) + 2(a1 b1 m21 m11 + a2 b2 m12 m22 ) |(a b − a b )| |(m11 m22 − m21 m12 ) ≤ 2 1 1 2 (a2 b1 + a1 b2 )(m11 m22 + m21 m12 ) = d (X, Y ) d (φ · A+ , φ · A− ) ≤ c (φ) d (X, Y ) . Above, we have used that 0 ≤ a1 , a2 , b1 , b2 and that m11 m22 + m21 m12 > 0 in order for the quotient above to be well-defined. This concludes the proof of part 1. For part 2, it is clear that c (φ) ≤ 1 from the definition. Now if φ is strictly positive we have that the projective action of φ on SD , φ·, is a continuous map on (SD , ∥ ∥) that take values in (SoD , ∥ ∥). Therefore we have that d (φ · X, φ · Y ) is a continuous map on (SD ×SD ) in to R where the former is equipped with the ∥ ∥-product topology. Since (SD × SD ) is compact (Tychonoff’s theorem) we have that F(X, Y ) = d (φ · X, φ · Y ) is a continuous map taking values strictly less than 1 for each X, Y ∈ SD × SD (due to proposition 2.6.1) with SD × SD compact and therefore we must have that c (φ) < 1. Part 3 of the lemma follows direct from proposition 2.6.2. Because of φ · X ∈ SoD and φ · Y ∈ SD \ SoD we have from proposition 2.6.2 that d (φ · X, φ · Y ) = 1. Now to prove part 4 note that for maps with no quantum states in their kernels we must have that (φ ◦ ψ) · X = φ · (ψ · X). Thus d (φ ◦ ψ · X, φ ◦ ψ · Y ) ≤ c (φ) d (ψ · X, φ · Y ) ≤ c (φ) c (ψ) d (X, Y ) which yields the desired result. Finally to prove part 5 first we claim that tr [AX] tr [A′ Y ] ( ) m(X, Y )m(Y , X) = inf : A, A′ ∈ SoD tr [A′ X] tr [AY ] 38 proof of which follows similar to part 6 of proposition 2.6.1, from which we get that tr [φ∗ (A)X] tr [[φ∗ (A′ )Y ] ( ) ′ o m(φ · X, φ · Y )m(φ · Y , φ · X) = inf : A, A ∈ SD . tr [φ∗ (A′ )X] tr [φ∗ (A)Y ] But we also have that tr [B(φ∗ · A)] tr [B′ (φ∗ · A′ )] ( ) ∗ ∗ ′ ∗ ′ ∗ ′ o m(φ · A, φ · A )m(φ · A , φ · A) = inf : B, B ∈ SD tr [B′ (φ∗ · A)] tr [B(φ∗ · A′ )] tr [B(φ∗ (A))] tr [B′ (φ∗ (A′ ))] ( ) ′ o = inf : B, B ∈ SD . tr [B′ (φ∗ (A))] tr [B(φ∗ (A′ ))] Therefore we have that tr [φ∗ (A)X] tr [[φ∗ (A′ )Y ] m(φ∗ · A, φ∗ · A′ )m(φ∗ · A′ , φ∗ · A) ≤ , tr [φ∗ (A′ )X] tr [φ∗ (A)Y ] and thus m(φ · X, φ · Y )m(φ · Y , φ · X) ≥ inf m(φ∗ · A, φ∗ · A′ )m(φ · Y , φ · X) : A, A′ ∈ SoD .  Taking infimum over X, Y ∈ SoD we see that m(φ∗ · A, φ∗ · A′ )m(φ · Y , φ · X) .  inf o {m(φ · X, φ · Y )m(φ · Y , φ · X)} ≥ inf X,Y ∈SD A,A′ ∈SoD Replacing φ by φ∗ and using that (φ∗ )∗ = φ we conclude the proof. Remark 2.8.2. If φ is a map such that ker(φ) ∩ SD = ∅ = ker(φ∗ ) ∩ SD , then c (φ) < 1 if and only if φ is strictly positive. We see from the lemma above that φ is strictly positive then c (φ) < 1. Now if φ is not strictly positive there is some X ∈ SD such that φ · X is positive semi-definite but not in SoD . Now as φ∗ has no positive-semi definite matrices in its kernel we must have from proposition 2.7.2, that φ(Y ) ∈ SoD whenever Y ∈ SoD thus for any such Y we have d (φ · X, φ · Y ) = 1. Thus c (φ) = 1 if φ is not strictly positive. We now present a proposition that will be useful in chapter 6. Proposition 2.8.3. Let ψ, φ ∈ L(MD ). Suppose that ψ is a positive map and φ is a strictly positive map with c (φ) ≤ r < 1. If ψ is non-transient, then for any A, B ∈ SD we have     2 1 ln ψ φ · A − ln ψ φ · B ≤ c(φ) ln . r 1−r 39 Proof. From proposition 2.6.1 we have that ( ) tr [XA] m(A, B) = min : X ∈ SD and tr [XA] , 0 . tr [XB] Since h i ψ(φ · A) tr [ψ ∗ (I) φ · A] tr ψ ∗ ( D1 I) φ · A = = h i, ψ(φ · B) tr [ψ ∗ (I) φ · B] tr ψ ∗ ( D1 I) φ · B we see that ψ(φ · A) 1 m(φ · A, φ · B) ≤ ≤ . ψ(φ · B) m(φ · B, φ · A) Since φ · A, φ · B are positive definite (because φ is strictly positive), the various terms appearing in this inequality are all finite and non-zero. Taking logarithms yields     ln ψ φ · A − ln ψ φ · B ≤ − ln m(φ · A, φ · B) − ln m(φ · B, φ · A) 1 + d (φ · A, φ · B) 1 + c (φ) ≤ ln ≤ ln , 1 − d (φ · A, φ · B) 1 − c (φ) where we have used the definition eq. 2.6.2 of d ( · , · ) and proposition 2.6.2 to obtain d (φ · A, φ · B) ≤ c(φ). Now for x ∈ [0, 1) we have 1+x 1 ≤ . 1−x (1 − x)2 As x = c (φ) ∈ (0, 1) (since φ is strictly positive) we have that     1 ln ψ φ · A − ln ψ φ · B ≤ 2 ln . 1 − c (φ) Now consider the convex function f (x) = ln 1/(1−x) for x ∈ [0, 1). Since f is convex and f (0) = 0, we have f (tr) ≤ tf (r) for any t, r ∈ [0, 1). Hence, f (λ) ≤ f (r)λ/r for any λ ∈ [0, r]. Thus whenever c(φ) ≤ r we have     2 1 ln ψ φ · A − ln ψ φ · B ≤ c(φ) ln . r 1−r 40 CHAPTER 3 DISCRETE PARAMETER QUANTUM PROCESSES 3.1 Introduction In this chapter we discuss asymptotic results for the behaviour of stationary quantum processes and other special types of stationary processes such as ergodic and IID pro- cesses. In general terms, a quantum process is a stochastic process taking values in quan- tum channels that is obtained by iterative application of sequence of quantum channel- valued random variables. Let (Ω, F , P) be a probability space and let φn : Ω → L(MD ) be a random variable taking values in quantum channels. Then a quantum process obtained by the sequence (φn )n∈N is the process, Φ (n) where Φ (n) : Ω → L(MD ) Φ (n) (ω) = φn (ω) ◦ . . . ◦ φ1 (ω) The term “stationary”, “ergodic”, or “IID” will refer to properties of the sequence (φn )n∈N that describe the process Φ (n) , not necessarily to properties of the process Φ (n) . In this the- sis we shall relax the property of φn being quantum channel-valued to be just completely positive maps or even just positive maps for the reasons explained in section 2.5 and section 2.7. A map φ in L(MD ) is called non-destructive if ker (φ) ∩ SD = ∅. A map φ in L(MD ) is called non-transient if ker (φ∗ ) ∩ SD = ∅. In this section we consider random quantum operations (i.e random maps defined on some probability space (Ω, F , P) taking values in completely positive operators in L(MD )) that are non-destructive and non-transient with probability 1. The use of this assumption will be clear with the following lemma for the contraction coefficient c defined for φ ∈ L(MD ): c (φ) = sup{d (φ · A, φ · B) : A, B ∈ SD }. We denote by P M the set of all positive maps, P M o the set of strictly positive maps, K the set of all non-destructive map and K ∗ the set of all non-transient maps. We consider 41 the subset P M ∩ K with the subset topology then we have the following lemma. Lemma 3.1.1. Consider the map c : P M ∩ K → [0, 1] given by c (φ) = sup{d (φ · A, φ · B) : A, B ∈ SD }. Then for φ ∈ K ∩ P M the following are true about the continuity of c at φ. 1. If φ is a strict positive map then c is continuous at φ. 2. If φ is not strictly positive but ker(φ∗ ) ∩ SD = ∅ then c is continuous at φ. Remark 3.1.2. Notice that L(MD ) is a normed space (and thus a metric space, where the metric is the induced metric of the norm) therefore L(MD ) is first countable. Thus the subset K ∩ P M topologized with the subspace topology is also first countable. For a map f : X → R where X is a first countable space we have that f is continuous at x0 ∈ X if and only if for all convergent sequences xn → x0 we have that lim f (xn ) = f (x0 ). (3.1.1) n→∞ We refer to any standard textbook on topology for these results [43, 22]. Proof. Let {φn }n∈N be a sequence in K ∩ P M such that φn converges to φ ∈ K ∩ P M (in operator norm). Since all φn and φ are in K we have that for all n ∈ N and X ∈ SD , φ · X and φn · X are defined. Now ||φn − φ|| → 0 implies that for all X ∈ SD , ||φn (X) − φ(X)|| → 0. Thus 0 < φ(X) /2 ≤ ||φn (X)|| eventually. Therefore for sufficiently large n we have φn (X) φ(X) 1 φ(X)( φn (X) − φ(X) ) − ≤ ||φn (X) − φ(X)|| + (3.1.2) φn (X) φ(X) φn (X) φn (X) φ(X) 2 2 ≤ φn (X) − φ(X) + φn (X) − φ(X) . (3.1.3) φ(X) φ(X) Taking limits we obtain that lim φn · X − φ · X → 0 for all X ∈ SD . (3.1.4) n→∞ Now since φn · X − φ · X converges to 0, we have by proposition 2.6.2 for any X ∈ SD and for any Y ∈ SoD , lim d (φn · X, Y ) = d (φ · X, Y ) . (3.1.5) n→∞ 42 Now we consider first of the two cases and assume that φ is strictly positive. Since φ is strictly positive, for all X ∈ SD , we have that φ(X) ∈ SoD . Therefore by 3.1.5 we have that lim d (φn · X, φ · X) = d (φ · X, φ · X) = 0 ∀X ∈ SD . (3.1.6) n→∞ Now for X, Y ∈ SD we have that d (φ · X, φ · Y ) ≤ d (φ · X, φn · X) + d (φn · Y , φ · Y ) + c (φn ) (3.1.7) because c(φn ) ≥ d (φn · X, φn · Y ). This gives us that lim inf c(φn ) ≥ d (φ · X, φ · Y ) for all X, Y ∈ SD . (3.1.8) n→∞ Hence, lim infn→∞ c(φn ) ≥ c(φ). On the other hand let n ∈ N and X, Y ∈ SD . Then we have that d (φn · X, φn · Y ) ≤ d (φn · X, φ · X) + d (φ · X, φ · Y ) + d (φ · Y , φn · Y ) (3.1.9) ≤ d (φn · X, φ · X) + c(φ) + d (φ · Y , φn · Y ) (3.1.10) Since above holds for all X, Y ∈ SD we have that lim sup c(φn ) ≤ c(φ). (3.1.11) n→∞ This completes the proof when φ is strictly positive. Now we consider the case where φ is positive but not strictly positive with the prop- erty that ker φ∗ ∩ SD = ∅. For such maps we have that there exists Z ∈ POSD such that φ(Z) is strictly positive definite (see proposition 2.7.3). Furthermore as φ is not a strictly positive map we have that there exists X0 ∈ SD such that φ(X0 ) ∈ POSD \ POSoD . Therefore using proposition 2.6.2 we have that 1 ≥ c(φ) ≥ d (φ · X0 , φ · Z) = 1. (3.1.12) Thus, c(φ) = 1 and we have that c(φn ) ≥ d (φn · Z, φn · X0 ) ≥ d (φ · Z, φn · X0 ) − d (φ · Z, φn · Z) (3.1.13) 43 Now using the fact that φ · Z ∈ SoD together with convergence in 3.1.5 we have, by taking limits that, lim inf c(φn ) ≥ d (φ · Z, φ · X0 ) = 1. (3.1.14) n→∞ The last equality comes form proposition 2.6.2. This proves the continuity of c at φ for the second case as we automatically have that lim supn→∞ c(φn ) ≤ 1. 3.2 Stationary Quantum Processes 3.2.1 Definition and Assumptions We note from proposition 2.2.4 it is reasonable to assume that a stationary stochastic process is constructed by translating a measurable map by a measure preserving trans- formation. We shall use this to our advantage in defining a stationary stochastic process. Definition 3.2.1 (Stationary Quantum Process). Let (φn , n ≥ 0) be a stationary sequence of positive map valued random variables defined on a probability space (Ω, F , P, θ) where θ is a measure preserving transformation of Ω with φn = φ0 ◦ θ n . For each n ∈ N define: Φ (n) = φn (ω) ◦ . . . ◦ φ1 (ω) (3.2.1) and for each m < n define (n) Φm = φn (ω) ◦ . . . ◦ φm (ω) (3.2.2) The sequence Φ (n) shall be called a stationary quantum process. We shall often omit ω from our notation and will simply write φn ◦ . . . φ1 or φn ◦ . . . φm . (Φ (n) )n∈N is called the (stationary) quantum process obtained by the stationary sequence (φn , n ≥ 1). It is important to notice that the sequence (Φ (n) )n∈N itself is not stationary. We shall assume the following: Assumption A1. There exists E ∈ F such that P(E) = 1 and for all ω ∈ Ω, ker(φ1 (ω)) ∩ SD = ker(φ1∗ (ω)) ∩ SD = ∅. 44 Remark 3.2.2. Since (φn )n∈N is stationary we have that under assumption 1, for all n ∈ N En ∈ F exists with probability 1 such that for any ω ∈ En which φn (ω) is both non-destructive and non-transient. We shall then take ∩n∈N En , which is a probability 1 event, then all maps φn will be non-destructive and non-transient with probability 1. In most proofs that follow we shall restrict our attention to the probability 1 event in which all maps φn are non-destrictive and non-transient. For n, m ∈ N ∪ {0} with m < n let    (n) cm,n := ln c Φm+1 , (3.2.3) where c (·) is defined as in 2.8.1. Under assumption 1 (and remark 3.2.2) we see that the for each m < n, cm,n defined above is almost surely measurable and therefore equal to a measurable function almost surely. We therefore extend the definition of cn,m so that cn,m is measurable in the usual sense. Lemma 3.2.3. For each m < n we have that cm,n : Ω → R is almost surely measurable. Proof. From lemma 3.1.1 we see that for any measurable φ : Ω → L(MD ) with ker (φ) ∩ SD = ker(φ∗ ) ∩ SD = ∅ we must have that ω 7→ ln c (φ(ω)) is almost surely measurable (equal to a measurable function almost everywhere). This is because for any Borel set B ∈ R we have that {ω : ln c (φ(ω)) ∈ B} ∩ (Ω \ N ) = {ω ∈ Ω \ N : ln c (φ(ω)) ∈ B, }. Here N is an event with probability 0 and outside N we have that ker φ(ω) ∩ SD = ker φ∗ (ω) ∩ SD = ∅. Now as ln is a continuous function and as c (·) is continuous func- tion on P M ∩ K ∩ K ∗ we have that {ω ∈ Ω \ N : ln c (φ(ω)) ∈ B} = {ω ∈ Ω \ N : φ(ω) ∈ U ∩ P M ∩ K ∩ K ∗ .} (3.2.4) where U is some Borel set in R. Now as φ is measurable we have from 3.2.4 that ln c (φ) is almost surely measurable. The rest of the proof follows by the following observation 45 on the sequence of maps:   ω 7→ φn (ω), φn−1 (ω), . . . , φm (ω) 7→ φn ◦ . . . φm . We have that first map is (⊗ni=m B, F )-measurable where B denotes the Borel σ -algebra on L(MD ) and ⊗ni=m B denotes the product σ -algebra. The second map is (B, B ( ni=m L(MD )))- Q measurable where B( ni=m L(MD )) is the Borel σ -algebra on ni=m L(MD ). Since L(MD ) is Q Q second countable we have that ⊗ni=m B = B( ni=m L(MD )) thus the above sequence is (B, F ) Q measurable. Therefore we have that for each m < n the map ω 7→ cm,n (ω) is almost surely measurable. Now we have that (cm,n )m N we have that c Φ0 (ω) < eMn . Now 47   (n) taking limit n → ∞ we have that limn→∞ c Φ0 (ω) = 0, as M < 0. Now for the case ξ(ω) ∈   (n) (−∞, 0) we have −(1/2) ln κ(ω) > 0. Then ln c Φ0 (ω) < n[ln κ(ω) − (1/2) ln κ(ω)] eventu-   (n) ally for large n. Thus for n large enough we have that c Φ0 (ω) ≤ exp{{(n/2) ln κ(ω)}}.   (n) Taking limit n → ∞ we see that c Φ0 (ω) → 0. 3.2.2 Limiting Results for the Stationary Case in Forward Time Let (φn , n ≥ 1) be a stationary sequence of positive map valued random variable sat- (n) (n) isfying assumption A1 for n > m define Φm = φn ◦ . . . φm . Now assume that Φ1 := Φ (n) satisfies assumption A2, then Φ (n) becomes eventually strictly positive (w.p. 1). Therefore it follows from [23], which is a generalization of the classical Perron-Frobenius theorem ([61, 27]) to C ∗ algebras, that eventually there exists unique (up to scaling) strictly posi- tive matrices Ln , Rn such that Φ (n) (Rn ) = Λn Rn and Φ (n)∗ (Ln ) = Λn Ln (3.2.7) where Λn is the spectral radius of Φ (n) (and hence the spectral radius of Φ (n)∗ ). That is, almost surely for ω there exists Nω such that for all n ≥ Nω , there exist positive matrices Rn , Ln such that 3.2.7 holds. We normalize Ln and Rn so that Ln , Rn ∈ SoD . The next results show that Ln converges as n → ∞. Theorem 3.2.6. Let (Φ (n) )n∈N be a stationary quantum process as defined in 3.2.1 with as- sumptions A1 and A2. Let Φ (n) and Ln (normalized so that Ln ∈ SoD ) be as in equation 3.2.7, Then there exists a stationary sequence (Zn )n∈N of random matrices such that 1. Z1 ∈ SoD almost surely and limn→∞ Ln converges (in trace norm) to Z1 almost surely. 2. For n ∈ N we have that φn∗ · Zn+1 = Zn almost surely. 3. For any k ∈ N and for any X ∈ SD we have that Zk is almost surely the norm limit of (φk∗ ◦ φk+1 ∗ ◦ . . . φn∗ ) · X as n → ∞.     4. For any Y ∈ SD and k < n ∈ N we have that d (φk∗ ◦ . . . ◦ φn∗ ) · Y , Zk ≤ c φk∗ ◦ . . . ◦ φn∗ almost surely. 48 Proof. Throughout this proof we restrict our attention on to the probability 1 event in which remark 3.2.2 and assumption A2 holds. For n ∈ N define Bn (ω) = Φ (n)∗ (ω) · SD . By assumption A2 we have that a.s. for ω Bn (ω) ∈ SoD eventually. That is almost surely for ω ∈ Ω there exists nω such that for all n ≥ nω , Bn (ω) ∈ SoD . From proposition 2.6.2 part (6) we have that for n ≥ nω we have that Bn (ω) is compact with respect to the d-metric topology (where d is the metric defined in 2.6.2). This is because Bn is compact in the ∥ ∥ topology for every n and from proposition 2.6.2 we have that (Sod , d) and (SoD , ∥ ∥) are homeomorphic. We also have that Bn+1 ⊆ Bn . Therefore we must have that ∩n≥nω Bn (ω) = ∩n∈N Bn (ω) is non-empty. We also have that       diam(Bn (ω)) = sup{d Φ (n)∗ · X, Φ (n)∗ · Y : X, Y ∈ SD } ≤ c Φ (n)∗ = c Φ (n) . (3.2.8)   But from proposition 3.2.5 we have that limn→∞ c Φ (n) = 0. Therefore we have that ∩n∈N Bn only has one element in the intersection. Let {Z1 (ω)} = ∩n∈N Bn (ω). By assump- tion A2 we have that Z1 (ω) ∈ SoD almost surely. Now we prove that limn→∞ Ln = Z1 almost surely. We first observe that Ln ∈ Bn even- tually. Therefore we have that d (Ln , Z1 ) → 0 as n → ∞. Then by proposition 2.6.2 part(3) we have that Ln → Z1 a.s. in trace norm too. Now we shall define Zn = Z1 ◦ θ n−1 for all n ≥ 2. By definition of Z1 we have that for   any X ∈ SD , limn→∞ [d (φ1∗ ◦ . . . ◦ φn∗ ) · X, Z1 → 0 a.s. that is   P ω : lim d ((φ1∗ ◦ . . . ◦ φn∗ ) · X, Z1 ) = 0 = 1. n→∞ Then as θ is measure preserving we must have that   P ω : lim d ((φ2∗ ◦ . . . ◦ φn+1 ∗ ) · X, Z1 ◦ θ) = 0 = 1. n→∞ Similarly we have that     P ω : lim d ∗ (φn+1 ∗ ◦ . . . ◦ φn+k ) · X, Z1 ◦ θ n =0 = 1. k→∞ 49 Therefore from proposition 2.6.2 we have that Zn is almost surely the norm limit of (φn∗ ◦ ∗ . . . ◦ φn+k ) · X = φn∗ ((φn+1 ◦ . . . ◦ φn+k ) · X) as k → ∞, for any X ∈ SD . Therefore we have that Zn = φn∗ · Zn+1 almost surely. Now to prove the final part we have that for any Y ∈ SD       d (φk∗ ◦ . . . ◦ φn∗ ) · Y , Zk = d (φk∗ ◦ . . . ◦ φn∗ ) · Y , ((φk∗ ◦ . . . ◦ φn∗ )) · Zn+1 ≤ c (φk∗ ◦ . . . ◦ φn∗ ) . Theorem 3.2.7. Let (Φ (n) )n∈N be a stationary quantum process as defined in 3.2.1 with as- sumption A1 and A2. Then we have that there exists a sequence of (random) matrices (Zn , n ≥ 1) and a random map µ such that µ < 1 almost surely and we have φ1∗ ◦ . . . ◦ φn∗ · ρ0 − Zn ≤ Dµ µn (3.2.9) where Dµ is a random map such that Dµ < ∞ with probability one. Furthermore the matrices Zn satisfy the shift equations: ∗ Zn = φn+1 · Zn+1 . (3.2.10) Proof. Similar to previous proof, we shall work on the probability 1 event in which the remark 3.2.2 and assumption A2 hold. Then ω in that event we have from proposition 2.6.2 and lemma 3.2.6 that   φ1∗ ◦ . . . ◦ φn∗ · ρ0 − Zn ≤ 2c Φ (n) . Let µ(ω) = (κ(ω) + 1)/2 ∈ (κ(ω), 1) then µ is measurable. We shall now define another measurable function C that is finite almost surely. Let C(ω) = inf{n ∈ N : c(φ(n+k) (ω)) − (µ(ω))n+k < 0 for all k ≥ 0}. The we have that C : Ω → R is indeed a measurable map as for any a ∈ R we have ⌊a⌋ n [  o {ω : C(ω) ≤ a} = {ω : C(ω) ≤ ⌊a⌋} = ω : c φ(n+k) (ω) − (µ((ω))n+k < 0 for all k ≥ 0 n=1 [⌊a⌋ \ n   o = ω : c φ(n+k) (ω) − (µ((ω))n+k < 0 . n=1 k≥0 50   Furthermore from theorem 3.2.4 we have that almost surely, (c φ(n) (ω) )1/n → κ as n → ∞. Thus almost surely for ω ∈ Ω there is some N (ω) ∈ N such that for all n ≥ N (ω) we will have c (φn (ω)) ≤ (µ(ω))n . Hence C(ω) ≤ N (ω) and thus C is almost surely finite. Now further restricting the probability space to where C is finite, if necessary, we have that for all n ≥ C(ω)   c Φ (n) (ω) ≤ µn (ω) ≤ (µ(ω))n (µ(ω))−C(ω) where the last inequality uses that µ < 1 and thus (µ(ω))−C(ω) ≥ 1. But for 1 ≤ n < C(ω) we have that (µ(ω)n ≥ (µ(ω))C(ω) and thus   c Φ (n) (ω) ≤ 1 ≤ (µ(ω))n (µ(ω))−C(ω) . Therefore we have that   c Φ (n) (ω) ≤ (µ(ω))n (µ(ω))−C(ω) for all n ∈ N. This gives us that φ1∗ ◦ . . . ◦ φn∗ · ρ0 − Zn ≤ 2(µ(ω))−C(ω) (µ(ω))n . We finish the proof by choosing D(ω) = 2(µ(ω))−C(ω) which is almost surely finite (because µ ∈ (0, 1) and C is almost surely finite). We also have that D = 2e−C ln µ therefore D is  measurable as C and ln µ measurable, so C ln µ is measurable and therefore exp −C ln µ is measurable. 3.2.3 Double-sided Stationary Processes: Definitions and Assumptions From lemma 2.2.5 we see that any one-sided stationary sequence can be embedded in a two-sided stationary sequence with same distribution in forward time. Therefore it is natural to consider stationary sequence that are two-sided. Furthermore from lemma 2.2.6 we see that it is natural to assume that the two sided sequence is constructed by shifting one map by an invertible measure preserving transformation. Using these obser- vations we define a two-sided stationary quantum process as follows: 51 Definition 3.2.8. Let (φn )n∈Z be a stationary (two-sided) sequence of positive maps defined on a common probability space (Ω, F , P, θ) where θ is an invertible measure preserving trans- formation on Ω with φn = φ0 ◦ θ n for n ∈ Z. For each n ∈ Z define   φn (ω) ◦ . . . ◦ φ1 (ω) if n > 0,        Φ (n) =   (3.2.11)   φ0 (ω) if n = 0,     φ−1 (ω) ◦ . . . ◦ φ−n (ω) if n < 0.   The process Φ (n) shall be called a double-sided stationary process. We note that we shall often omit ω from notation for ease. Similar to the one-sided case we note that the sequence (φn , n ∈ Z) describing the process (φ(n) )n∈Z is stationary and (Φ (n) )n∈Z itself is not. A major detail in previous section was the (eventual) existence of Perron-Frobenius eigenmatrices which was a consequence of the process becoming eventually strictly pos- itive as such we assume the following assumptions. Assumption C1. Assume that there exists an even E ∈ F such that P(E) = 1 and for all ω ∈ E, φ0 (ω) is both non-destructive and non-transient. Assumption C2. We assume that almost surely for all ω there exists N1,ω and N2,ω such that φ(N1,ω ) (ω) and φ−(N2,ω ) (ω) are strictly positive. Lemma 3.2.9. Let (Φ (n) )n∈Z be a two-sided stationary process defined as in 3.2.8 under as- sumptions C1 and C2 we have that almost for every ω ∈ Ω there is some nω ∈ SD such that for all |n| ≥ nω , Φ (n) is strictly positive. Proof. We restrict our attention to the probability 1 event in which assumption C2 holds and the kernels of all maps φn and φn∗ (for n ∈ Z) have trivial intersection with SD . Even- tual strict positiveness in the forward time follows from part (1) of proposition 3.2.5. To prove the result in reverse direction we use the kernel proprieties. By assumption C2 we 52 have almost for every ω there is a Nω ∈ N such that Φ (−Nω ) is strictly positive. Now let k ≤ 0 then we have that Φ (−Nω −k) = φ−1 ◦ . . . φ−Nω ◦ φ−Nω −1 ◦ . . . φ−Nω −k . Now as φ−1 ◦ . . . φ−Nω is strictly positive and as φ−Nω −1 ◦ . . . φ−Nω −k has trivial kernel on SD (by assumption C1), we must have that Φ (−Nω −k) maps any X ∈ SD into SoD , i.e. Φ (−Nω −k) is strictly positive. 3.2.4 Limiting Results for Two-sided Stationary Processes With the support of lemma 3.2.9 we define the following random variable, τ, and the previous lemma shows us that τ < ∞ almost surely τ : Ω → N ∪ {0} (3.2.12) (|n+k|) τ(ω) = inf{n ≥ 0 : φ (ω) is strictly positive ∀k ≥ 0}. Since w.p.1. the process Φ (n) becomes eventually strictly positive (in both directions) we have from [23] almost for every ω left and right Perron-Frobenius eigenmatrices Ln and Rn exist uniquely (upto scaling) for sufficiently large |n| such that Φ (n) (ω)(Rn (ω)) = Λn (ω)Rn (ω) and Φ (n)∗ (Ln (ω)) = Λn (ω)Ln (ω). (3.2.13) We normalize Ln and Rn so that they are of trace 1. To study the limiting results in the backward direction we shall introduce new notations. Let n ∈ N and define ψn = ϕ−n ∗ and Ψ (n) = ψn ◦ . . . ◦ ψ1 . (3.2.14) Note that Ψ (n)∗ = Φ (−n) . We see that (Ψ (n) )n∈N is almost surely eventually strictly positive by lemma 3.2.9. It follows from [23], that (eventually for large positive n) there exist unique (up to scaling) strictly positive matrices L′n , and R′n such that Ψ (n) (R′n ) = Λn R′n and Ψ (n)∗ (L′n ) = Λn L′n . (3.2.15) 53 Here we are using that spectral radius of Ψ (n) is the same as spectral radius of Φ (−n)∗ and hence equivalent to the spectral radius of Φ (−n) . We normalize L′n and Rn so that L′n , R′n ∈ SoD . The following lemmas are similar to the theorem 3.2.4, proposition 3.2.5 and lemma 3.2.6 and thus state them without proofs. Lemma 3.2.10. Let Ψ (n) be as in 3.2.14. Assume that the original sequence φn satisfies the assumptions C1 and C2, then there exists a (random) κ′ ∈ [0, 1) almost surely such that a.s 1   ln κ′ = lim ln c Ψ (n) . (3.2.16) n→∞ n   Furthermore we have that c Ψ (n) → 0 almost surely as n → ∞. Lemma 3.2.11. Let Ψ (n) be as in 3.2.14. Assume that the original sequence φn satisfies the assumptions C1 and C2. Let L′n (normalized so that L′n ∈ SoD ) be as in equation 3.2.15. Then there exists a stationary sequence (Zn′ )n∈N of random matrices such that Zn′ = Z1′ ◦ θ −n+1 and 1. Z1′ ∈ SoD almost surely and limn→∞ L′n converges (in trace norm) to Z1′ almost surely. 2. For k ∈ N and for any X ∈ SD we have that almost surely Zk′ is the norm limit of (ψk∗ ◦ . . . ◦ ψn∗ ) · X as n → ∞. ′ 3. For n ∈ N we have that ψn∗ · Zn+1 = Zn′ .     4. For any Y ∈ SD and k < n ∈ N we have that d (ψk∗ ◦ . . . ◦ ψn∗ ) · Y , Zk ≤ c ψk∗ ◦ . . . ◦ ψn∗ . We now have the following corollary due to lemma 3.2.11 Corollary 3.2.12. Let Ln and Rn be as in equation 3.2.13. Then we have that a.s lim Rn = Z1′ , (3.2.17) n→−∞ Proof. By (up to scaling) uniqueness of Ln , Rn and L′n , R′n and also from the fact that all Ln , Rn , L′n , R′n are positive definite we have that (eventually) for n ≥ 0, L′n = R−n therefore a.s we have from 3.2.11 that limn→−∞ Rn = Z1′ Remark 3.2.13. From lemma 3.2.11 we see that the sequence (Zn′ ; n ≥ 1) has the property that ′ ′ ψn∗ · Zn+1 = Zn′ . Therefore we have that Zn′ = φ−n · Zn+1 . We shall re-index the sequence Zn′ , call 54 this Zn′′ so that Zn′′ = Z−n ′ for n < 0. We note that the indexing for Z ′′ now runs over negative n integers. Now the shift equation reads ′′ ′′ Z−n = φ−n · Z−n−1 for all n ≥ 1. (3.2.18) Since Zn′ = Z1′ ◦ θ −n+1 we have that Z”−n = Z−1 ′′ ◦ θ −n+1 , with the uses of this property we define Zn′′ for all n ∈ Z: Zn′′ = Z−1 ” ◦ θ n+1 . With the re-indexed Zn′′ with the extension for all integer indices we have the following theorem: Theorem 3.2.14. Let (Φ (n) )n∈Z be a stationary quantum process obtained from a two-sided stationary sequence of random positive maps as defined in 3.2.8 with assumption C1 and C2. Then there are two-sided stationary sequences of (random) matrices (Zn )n∈Z and (Zn′′ )n∈Z such ′′ that Zn = Z1 ◦ θ n−1 , Zn′′ = Z−1 ◦ θ n+1 and 1. Z1 ∈ SoD almost surely and limn→∞ Ln converges (in trace norm) to Z1 almost surely. 2. For n ∈ Z we have that φn∗ · Zn+1 = Zn almost surely. 3. For any Y ∈ SD and k < n ∈ Z we have that (φk∗ ◦ . . . ◦ φn∗ ) · Y converges (as n → ∞) to Zk (in trace norm) almost surely.     4. For any Y ∈ SD and k < n in Z we have that d (φk∗ ◦ . . . ◦ φn∗ ) · Y , Zk ≤ c φk∗ ◦ . . . ◦ φn∗ . 5. Z−1′′ ∈ SoD almost surely and limn→−∞ Rn converges (in trace norm) to Z−1 ′′ almost surely. 6. For k ∈ Z, Zk′′ is the almost sure (trace) norm limit (as n → ∞) of (φk ◦ . . . ◦ φ−n ) · X for any X ∈ SD . 7. For k ∈ Z we have that Zk′′ = φk · Zk−1 ′′ almost surely. 8. For any Y ∈ SD and m < n in Z we have that d ((φn ◦ . . . ◦ φm ) · Y , Zn′′ ) ≤ c (φn ◦ . . . ◦ φm ). Proof. Existence of the Z1 for n ≥ 1 is directly from theorem 3.2.6. For n ∈ Z we define Zn = Z1 ◦θ n−1 . Then (Zn )n∈Z is clearly a doubly infinite stationary sequence. Furthermore we have that Z1 is the norm limit of (φ1∗ ◦ . . . φn∗ ) · X for any X ∈ SD , almost surely. Now as 55 θ and θ −1 are measure preserving we must have for any k ∈ Z, Z1 ◦ θ k is the norm limit ∗ ∗ of (φ1+k ◦ . . . φn+k ) · X for any state X, almost surely. And thus for any X ∈ SD Zk = lim (φk∗ ◦ . . . φn+k∗ ) · X = φk · lim (φk+1 ∗ ∗ ◦ . . . φn+k ) · X = φk · Zk+1 . n→∞ n→∞ Now for any Y ∈ SD we have that for k < n;       d (φk∗ ◦ . . . ◦ φn∗ ) · Y , Zk = d (φk∗ ◦ . . . ◦ φn∗ ) · Y , (φk∗ ◦ . . . ◦ φn∗ ) · Zn+1 ≤ c φk∗ ◦ . . . ◦ φn∗ . ′′ This proves the first 4 parts of the theorem. Now for the last 4 parts, let Z−1 = Z1′ be as ′′ in lemma 3.2.11. Then from the proof of 3.2.11 we have that Z−1 is the almost sure norm ′′ limit of (φ−1 ◦ . . . ◦ φ−n ) · X, for any X ∈ SD . Which also means that Rn → Z−1 as n → −∞. Also, as θ and θ −1 are measure preserving we have that for any k ∈ Z, Zk′′ := Z−1 ′′ ◦ θ k+1 is the almost sure norm limit of (φk ◦ . . . ◦ φ−n ) · X and therefore we have that Zk′′ = φk · Zk−1 ′′ . Rest of the proof is similar. Theorem 3.2.15. Let (Φ (n) )n∈Z be a stationary quantum process obtained from a two-sided stationary sequence of random positive maps on (Ω, F , P, θ) as defined in 3.2.8 with assump- tion C1 and C2. Then we have that there exists a random matrix valued map Z ′′ : Ω → MD and a random map µ such that µ < 1 almost surely and for Zn′′ = Z ′′ ◦ θ n we have φn ◦ . . . ◦ φ1 · ρ0 − Zn′′ ≤ Cµ µn for n ∈ N, (3.2.19) for any initial state ρ0 , where Cµ < ∞ with probability one. Furthermore the matrices Zn satisfy the shift equations: Zn′′ = φn · Zn−1 ′′ . (3.2.20) Proof. Let Zn′′ be as theorem 3.2.14 then we have for any ρo ∈ SD   d (φn ◦ . . . ◦ φ1 · ρ0 , Zn′′ ) = d ((φn ◦ . . . ◦ φ1 ) · ρ0 , (φn ◦ . . . φ1 ) · Z0′′ ) ≤ c Φ (n) . Then by proposition 2.6.2 we have that   φn ◦ . . . ◦ φn · ρ0 − Zn′′ ≤ 2c Φ (n) . The rest of the proof is similar to the proof of theorem 3.2.7. 56 Before we introduce our next theorem we shall prove a couple of lemmas that will be (n) useful later. We need few more notations. For m < n let Φm = φn ◦ . . . ◦ φm and let Pn,m for n, m ∈ Z be the rank one operator Pn,m (X) = tr [Zm X] Zn′′ , where Zm and Zn′′ are as in theorem 3.3.10 Lemma 3.2.16. For m < n ∈ Z and X ∈ SD we have that    (n) (n)  tr Φm (X)  Φm (X)     (n)  (n)  − Pn,m (X) ≤ 2c Φm 1 +    ≤ 4c Φm (3.2.21)      (n)∗ (n)∗ tr Φm (I)  tr Φm (I)  almost surely. Proof. Let Y ∈ SD . Then from theorem 3.2.14 we have that d ((φn ◦ . . . ◦ φm ) · Y , Zn′′ ) ≤ c (φn ◦ . . . ◦ φm ). Therefore from proposition 2.6.2 we have that   (n) (n) Φm · Y − Zn′′ ≤ 2c Φm almost surely. (3.2.22) Similarly we have that     (n)∗ (n)∗ (n) tr Φm · Y − tr [Zm ] ≤ Φm · Y − Zm ≤ 2c Φm almost surely. (3.2.23) Since the above inequality holds for any Y ∈ SD , in particular for Y = I/D we must have that, almost surely (n)∗ φm (I)   (n)   − Zm ≤ 2c Φm . (3.2.24) (n)∗ tr φm (I) Now for X ∈ SD we have that       (n) (n)∗ tr φm (X) tr φm (I)X  (n)∗  φm (I)X     − tr [Zm X] =   − tr [Zm X] = tr      − tr [Zm X] (n) (n)∗ (n)∗ tr φm (I) tr φm (I)  tr φm (I)   (n)∗ φm (I)   (n) ≤   − Zm · ∥X∥ ≤ 2c Φm almost surely. (3.2.25) (n)∗ tr φm (I) 57 Then from 3.2.25 we get that   (n) tr φm (X)   ′′ ′′ (n)   Zn − tr [Zm X] Zn ≤ 2c Φm almost surely. (3.2.26) (n) tr φm (I)    (n) (n)∗ Multiplying the inequity 3.2.23 by tr Φm (X)/tr Φm (I) and combining it with 3.2.25 we see that:    (n) (n)  tr Φm (X)  Φm (X)      (n)  (n)  − Pn,m (X) ≤ 2c Φm 1 +    ≤ 4c Φm    (n)∗ (n)∗ tr Φm (I)  tr Φm (I)        (n) (n)∗ (n)∗ Where the last inequality comes from tr Φm (X) = tr φm (I)X ≤ tr Φm (I) as ∥X∥ = 1. Now let us define the following measurable map 1   µ(ω) = max{κ(ω), κ′ (ω)} + 1 (3.2.27) 2 where κ and κ′ are as in 3.2.4 and 3.2.10 respectively. µ is indeed measurable as maximum of two measurable functions are measurable. Then we introduce a new random variable as follows N1 : Ω → N   (3.2.28) N1 (ω) = inf{n ∈ N : c φ(n+k) (ω) − (µ(ω))n+k < 0 for all k ≥ 0.}. We have that N1 is indeed a random variable, i.e a measurable map taking real values defined on Ω. Notice that from lemma 2.1.2 it is enough to show that the set {N1 ≤ a} is measurable for any a ∈ R. Indeed this is the case as [⌊a⌋ n   o {ω : N1 (ω) ≤ a} = {ω : N1 (ω) ≤ ⌊a⌋} = ω : c φ(n+k) (ω) − (µ(ω))n+k < 0 for all k ≥ 0 j=1 [ ⌊a⌋ \ n   o = ω : c φ(n+k) (ω) − (µ(ω))n+k < 0 j=1 k≥0 58 Similarly in the negative direction we have that N2 (ω) defined as   ∗ N2 (ω) = inf{n ≥ 1 : c φ−n−k ∗ ◦ . . . ◦ φ−1 (ω) − (µ(ω))n+k < 0 for all k ≥ 0} (3.2.29) is measurable. We summarize the discussion above as a lemma below. Lemma 3.2.17. Let µ, N1 and N2 be as in 3.2.27, 3.2.28 and 3.2.29 (resp.) then we have that 1. µ, N1 and N2 are measurable functions. 2. µ ∈ (0, 1) almost surely. 3. N1 , N2 are finite almost surely. Proof. The first 2 properties are discussed above. To prove that N1 < ∞ a.s. notice that from theorem 3.2.4 we have that a.s. for ω there is some Nω such that for all n ≥ Nω   c Φ (n) (ω) < (µ(ω))n , and thus N1 (ω) ≤ Nω a.s. Therefore N1 (ω) is almost surely finite. With lemma 3.2.10, result for N2 is similar. Equipped with the previous two lemmas and the random maps µ, N1 and N2 we present our next theorem which shows that for n − m large the operator φn ◦ . . . ◦ φm get asymptotically closer to a family of rank 1 operators. Theorem 3.2.18. There is a sequence of rank one operators Pn,m , a measurable map µ : Ω → (0, 1) and a measurable map D : Ω → R such that for each x ∈ Z we have that 1 (n)   Φm − Pn,m ≤ D(θ x )(µ(θ x ))n−m almost surely (3.2.30) (n) tr Φm (I) for all m ≤ x and n ≥ x with D finite almost surely and depends on µ. Proof. Let Pn,m (X) = tr [Zm X] Zn′′ where Zm and Zn′′ are as in theorem 3.2.14. By the defini- tion of the operator norm (see 2.4.8) it is sufficient to prove that for any X ∈ SD , we have: 59 almost surely, 1 (n)   Φm (X) − Pn,m (X) ≤ D(θ x )(µ(θ x ))n−m ||X||. (n) tr Φm (I) P4 P4 This is because for any X ∈ MD we have that X = i=1 ai Xi with Xi ∈ SD and i=1 |ai | ≤ 2∥X∥. Now from lemma 3.2.17 we see that almost for all ω there exists some N1 (ω) ∈   N such that for all n ≥ N1 , c Φ (n) (ω) ≤ µn (ω) ≤ (µ(ω))n (µ(ω))−N1 (ω) . Where the last inequality uses that µ < 1. Now as µ ∈ (κ, 1), we have for 0 ≤ n < N1 , (µ(ω))n ≥ (µ(ω))N1 (ω) .   Thus for n < N1 (ω) we have c Φ (n) (ω) ≤ 1 ≤ (µ(ω))n (µ(ω))−N1 (ω) . Therefore we have that   c Φ (n) (ω) ≤ (µ(ω))n (µ(ω))−N1 (ω) for all n ∈ N ∪ {0}. (3.2.31) On the other hand we also have from lemma 3.2.17 that almost surely for ω ∈ Ω, there   is some N2 (ω) such that for all n ≥ N2 (ω), c Φ −(n) (ω) ≤ (µ(ω))n . Since µ < 1 we have   that for all n ≥ N2 (ω), c Φ −(n) (ω) ≤ (µ(ω))n ≤ (µ(ω))n (µ(ω))−N2 (ω) . Similar to 3.2.31 we   also have for 0 ≤ n ≤ N2 (ω), as m < 1, (µ(ω))n ≥ (µ(ω))N2 (ω) and thus c Φ −(n) (ω) ≤ 1 ≤ (µ(ω))n (µ(ω))−N2 (ω) Thus we have that   c Φ −(n) (ω) ≤ (µ(ω))n (µ(ω))−N2 (ω) for all n ∈ N ∪ {0}. (3.2.32) Now let M be the random variable defined as: M(ω) = max{N1 (ω), N2 (ω)}. (3.2.33) By combining 3.2.31 and 3.2.32 we must have that   c Φ (n) (ω) ≤ (µ(ω))−M(ω) (µ(ω))n ∀ n ∈ N ∪ {0}, (3.2.34) and that   c Φ −(n) (ω) ≤ (µ(ω))−M(ω) (µ(ω))n ∀ n ∈ N ∪ {0} (3.2.35) holds almost surely. 60 We shall define C(ω) = (µ(ω))−M(ω) . Then we have that C is measurable because we have C = e−M ln µ . First let x ∈ Z and let m ≤ x < n ∈ Z. Then we have that n − x ≥ 1 and using 3.2.34 let E be the probability 1 event in which c (Φ n−x (ω)) ≤ C(ω)(µ(ω))n−x . Then as θ is a bi-measurable measure preserving transformation we have that θ −x (E) also is of probability 1. Thus   (n) c φx+1 (ω) ≤ C(θ x (ω))(µ(θ x (ω)))n−x w.p.1. (3.2.36) Similarly, if x = m we have that, as c (φ0 ) ≤ C(ω) w.p. 1 by 3.2.34. Then translating by θ x it must be the case that c (φx ) ≤ C(θ x (ω)) w.p.1. (3.2.37) If m < x we have that m − x ≤ −1 and by 3.2.35 we have that c (φ−1 ◦ . . . ◦ φm−x ) ≤ C(ω)(µ(ω))m−x with probability one. Then by translating we see that c (φx−1 ◦ . . . ◦ φm ) ≤ C(θ x (ω))(µ(θ x (ω)))m−x w.p.1. (3.2.38) Combining 3.2.36, 3.2.37 and 3.2.38 we see that for the case m ≤ x < n we have that    (n) c φx+1 c (φx ) ≤ (C(θ x (ω)))2 (µ(θ x (ω)))n−x if x = m,     (n)  c Φm ≤        (n) (x−1) c φx+1 c (φx ) c Φm ≤ (C(θ x (ω))2 (µ(θ x (ω)))n−x (µ(θ x (ω)))x−m if x > m.   (3.2.39) Similarly for the case m ≤ x = n we have that   (n)     c (φx ) ≤ C(θ x (ω)) if n = x = m, c Φm ≤  (3.2.40)    (x−1) ≤ C(θ x (ω))(µ(θ x (ω)))x−m  c (φx ) c Φm if x > m.   Since by lemma 3.2.16 we have that for any X ∈ SD (n)  (n)  Φm (X) Φ (X)     (n) m  (n)  − Pn,m (X) ≤ 2c Φm 1 + (n)∗  ≤ 4c Φm   (n)∗ tr Φm (I) Φm (I) we can now combine 3.2.39 and 3.2.40 to get the desired results with Dω,x,µ ≈ K max{C(ω), C(ω)2 } 61 is some almost surely finite random variable with an appropriate constant K. This con- cludes the proof with the discussion in the beginning of the proof. 3.3 Discrete Parameter Ergodic Quantum Processes In this section we consider a special case of stationary quantum processes that we dis- cussed earlier in this chapter. Here the sequence of random positive maps that describes the process is assumed not only to be stationary but also ergodic. 3.3.1 Definition and Assumption Definition 3.3.1. Let (Ω, F , P, θ) be a probability space where θ is an ergodic measure pre- serving transformation on Ω. Let φ0 : Ω → L(MD ) be a random variables taking values in positive operators. Then for each n ∈ N define φn (ω) = φ0 ◦ θ n (ω). (3.3.1) Now consider the quantum process obtained by the ergodic sequence (φn )n∈N defined as Φ (n) : Ω → L(MD ) Φ (n) (ω) = φn (ω) ◦ . . . ◦ φ1 (ω). The sequence Φ (n) is then called an ergodic quantum process. Similar to the stationary case we shall omit ω from notation for ease and we note that ergodicity is a property that pertains to the sequence (φn )n≥1 that describes the quantum process (Φ (n) )n∈N . Similar to the stationary case we shall assume the following: Assumption B1. There exists E ∈ F such that P(E) = 1 and for all ω ∈ E, ker(φ0 (ω)) ∩ SD = ker(φ0∗ (ω)) ∩ SD = ∅. Remark 3.3.2. Similar to the remark 3.2.2 assumption B1 implies the existence of a probability 1 event in which kernels of φn and φn∗ does not contain any matrices in SD for all n ∈ N. We shall often restrict our proof to such an event. 62 A main property of the quantum process that was useful for the proofs in the sta- tionary case was that the process became eventually strictly positive w.p. 1. This was guaranteed by the assumption that the process became strictly positive in finite time. However for the ergodic case we shall assume the existence of some N such that up to N composition is strictly positive with positive probability: Assumption B2. ∃N0 ∈ N such that P{ω : Φ (N0 ) (ω) : is strictly positive } > 0. We have under these assumptions the process does indeed become eventually strictly positive with probability 1. Lemma 3.3.3. For an ergodic quantum process defined as 3.3.1 we have if assumptions B1 and B2 hold then the the process becomes strictly positive eventually w.p. 1. that is the random variable τ below is finite a.s. τ(ω) = inf{n ≥ 1 : φ(n+k) (ω) is strictly positive ∀k ≥ 0}. (3.3.2) Proof. We restrict ω in this proof to the probability 1 event where remark 3.3.2 and as- S n o sumption B2 hold. Let X = k≥0 ω : Φ (N0 ) ◦ θ k (ω) is strictly positive . Then we have that θ −1 (X) ⊆ X and as θ is measure preserving we must have that P(θ −1 (X))) = P(X). Thus P(θ −1 X∆X) = 0. Therefore P(X) ∈ {0, 1} by ergodicity. Since there exists N0 such that Φ (N0 ) is strictly positive with positive probability we must have that P(X) = 1. Therefore a.s. for each ω there exists N0 < ∞ such that Φ (N0 ) (θ n (ω)) is a strictly positive map. Now we have that    (N0 ) (ω) if n = 0 Φ   (N0 +n)  Φ (ω) =   (3.3.3) Φ (N0 ) (θ n (ω)) ◦ Φ (n) (ω)  if n ≥ 1   Since ω is taken from a probability 1 event in which ker (φn (ω)) ∩ SD = ∅, for all n ∈ N we have that for any nonzero positive semi-definite matrix A, (Φ (n) (ω))(A) is non-zero pos- itive semi-definite and as Φ (N0 ) is strictly positive, Φ (N0 +n) (A) becomes positive definite almost surely. Now since N0 + n < ∞ we see that τ < ∞ a.s. This proves that the process 63 becomes strictly positive in finite time a.s.. Now to prove that once the process becomes strictly positive it must stay strictly positive there after we use proposition 2.7.2, which hold by assumption B1. Similar to the stationary case, under these assumptions we have that for n, m ∈ N∪{0}, with m < n, the family (cm,n , m < n) satisfy following 3 conditions where    (n) cm,n := ln c Φm+1 . (3.3.4) S’1. Whenever l < m < n, we have that cn,l ≤ cm,l + cm,n . S’2. The joint distribution of the processes (cm+1,t+1 )m r) = P(ξ ◦ θ > r) for any r ∈ R. Therefore [n o P(ξ , ξ ◦ θ) = P{ω : ξ(ω) < (ξ ◦ θ)(ω)} = P ω : ξ(ω) < r < (ξ ◦ θ)(ω) r∈Q X ≤ P{ω : ξ(ω) ≤ r < ξ ◦ θ(ω)} r∈Q X   ≤ P {ω : ξ ◦ θ > r} \ {ω : ξ > r} r∈Q = 0. (3.3.8) Thus ξ is essential θ-invariant whence ξ is a.s. a constant and must equal to E[ξ] a.s. 3.3.2 Limiting Results in Forward Time Similar to the stationary case we have that (eventually) there exist unique (up to scal- ing) strictly positive matrices Ln , Rn such that Φ (n) (RN ) = Λn Rn and Φ (n)∗ (LN ) = Λn Ln (3.3.9) That is, almost surely for ω there exists Nω such that for all n ≥ Nω , there exist positive matrices Rn , Ln such that 3.2.7 holds. We normalize Ln and Rn so that Ln , Rn ∈ SoD . We have the following corollary of theorem 3.2.6. Corollary 3.3.5. Let Φ (n) be an ergodic quantum process as defined in 3.3.1. Assume that B1 and B2 hold. Then let Φ (n)∗ and Ln be as in equation 3.3.9. Then as n → ∞, Ln converges a.s. to a limit Z1 , such that Z1 ∈ SoD a.s.. Furthermore for Zk := Z1 ◦ θ k−1 we have that; 1. For k ≥ 1, Zk is also positive definite a.s. 2. For any k ∈ N and for any X ∈ SD we have that Zk is almost surely the norm limit of (φk∗ ◦ φk+1 ∗ ◦ . . . φn∗ ) · X as n → ∞. 3. For k ∈ N, ϕk∗ · Zk+1 = Zk a.s.     4. For Y ∈ SD and k, n ∈ N, d (ϕk∗ ◦ . . . ϕn∗ ) · Y , Zk ≤ c ϕk∗ ◦ . . . ϕn∗ a.s. 65 3.3.3 Ergodic Quantum Processes from an Invertible Ergodic System As explained in section 2.3 it is natural to assume that the ergodic map is invertible. For such cases we define an ergodic quantum process similar to the previous case. How- ever we note that the assumptions B1 and B2 may be combined in to one assumption for this case as we shall see below. Definition 3.3.6. Let (Ω, F , P, θ) be a probability space where θ is a bi-measurable ergodic measure preserving transformation on Ω. Let φ0 : Ω → L(MD ) be a random variable taking values in positive operators then for each n ∈ Z define φn (ω) = φ0 ◦ θ n (ω). (3.3.10) Then consider the quantum process obtained by the ergodic sequence (φn )n∈N defined as Φ (n) : Ω → L(MD )   φn (ω) ◦ . . . ◦ φ1 (ω) for n ≥ 1,        Φ (n) (ω) =     φ0 (ω) for n = 0,     φ−1 (ω) ◦ . . . ◦ φ−n (ω) for n ≤ −1.   The sequence Φ (n) is then called an ergodic quantum process. Similar to the previous case we start by assuming the following: Assumption B’1. There exists E ∈ F such that P(E) = 1 and for all ω ∈ Ω, ker(φ0 (ω))∩SD = ker(φ0∗ (ω)) ∩ SD = ∅. Assumption B’2. ∃N0 ∈ N such that P{ω : Φ (N0 ) (ω) : is strictly positive } > 0. Similar to lemma 3.3.3 we first show that these two assumptions are enough to guar- antee that the process Φ (n) becomes strictly positive eventually in both directions w.p 1. The result in forward time is just lemma 3.3.3. 66 Lemma 3.3.7. For an ergodic quantum process on an invertible ergodic system defined as 3.3.1 we have if assumptions B’1 and B’2 hold then the the process becomes strictly positive eventu- ally w.p. 1. that is the random variable τ below is finite a.s. τ(ω) = inf{n ≥ 1 : φ|n+k| (ω) is strictly positive ∀k ≥ 0}. (3.3.11) Proof. From 3.3.3 we know that almost for all ω there exists nω such that for Φ (nω ) (ω) is strictly positive. Thus the random variable τ defined below is finite w.p.1. σ (ω) = inf{n ≥ 1 : φ(n) (ω) is strictly positive } Now for n ∈ N define In = {ω ∈ Ω : σ (ω) = n}. Then we have that   X ∞ 1 = PA = P ∪n≥1 In ≤ PIn . (3.3.12) n=1 Thus there exists M0 ≥ 1 such that P(IM0 ) > 0. Now consider θ M0 +1 (IM0 ) we must have that P(θ M0 +1 (IM0 )) > 0, as θ −1 is measure preserving. Therefore we have that: There is some M0 such that the event E = {ω : φ−1 ◦ . . . φ−M0 is strictly positive} has positive probability. Now let X = S −k k≥0 {(φ−1 ◦. . . φ−M0 )◦θ is strictly positive }. Then we have that θ(X) ⊆ X but as θ −1 is also ergodic we must have that P(X) = 0 or 1 but E ⊆ X already has positive probability thus P(X) = 1. Therefore for almost all ω there is some k ≥ 0 such that φ−1−k ◦ . . . φ◦−M0 −k is strictly positive then we have Φ (−M0 −k) = φ−1 ◦ . . . φ−k ◦ φ−1−k ◦ . . . φ◦−M0 −k is strictly positive because φ−1−k ◦ . . . φ◦−M0 −k is strictly positive and by the kernel property of (φ−1 ◦ . . . ◦ φ−k )∗ in assumption B′ 2 together with proposition 2.7.2. Then we have that Φ (−M0 −k−l) is strictly positive for l ≥ 0 by the kernel properties of (φ−M0 −k−1 ◦ . . . φ−m0 −k−l ) in assumption B’2. In the invertible ergodic case we have that the assumptions B’1 and B’2 are equivalent to the assumption that: 67 Assumption B’. The random time τ : Ω → R where τ(ω) = inf{n ≥ 0 : Φ |n+k| is strictly positive ∀k ≥ 0} is almost surely finite. Lemma 3.3.8. In an invertible system B’1 and B’2 hold if and only if B’ holds. Proof. We already saw that B’1 and B’2 implies B’ in the previous lemma. Now we prove the other direction. If τ < ∞ almost surely we have that X 1 = P(τ < ∞) = P(∪n≥0 In ) ≤ P(In ) n∈N where In = {ω : τ(ω) = n}. Then there must exist N0 such that P(IN0 > 0. i.e P({ω : τ(ω) = N0 }) > 0. Thus at least one of the sets A1 = {ω : φN0 is strictly positive } or A2 = {ω : φ−N0 is strictly positive } must have positive probability. If A1 has positive probability we have B’2 if A2 has positive probability we shift A2 by θ −N0 −1 to get B’2. So we have that B’⇒ B’2. Now let B = {ω : ker(φ0 ) ∩ SD , ∅} then θ −1 (B) = {ω : ker(φ1 ) ∩ SD , ∅}. Now let ω ∈ θ −1 (B) then we must have that ω < {ω : τ(ω) < ∞}. Thus θ −1 (B) has 0 probability and so would B. Now let ω be such that τ(ω) < ∞ we then have that there is some nω such that Φ (n) (ω) is strictly positive for all n ≥ nω . Then for any X, Y ∈ POSD we have 0 < ⟨X, (Φ (n) (ω))(Y )⟩ = ⟨(φn∗ (ω))(X), (Φ (n−1) (ω))(Y )⟩ for all n ≥ nω . Therefore it must be the case that φn∗ does not contain any quantum states in its kernel. That is almost for all ω there is some nω such that for all n ≥ nω ker (φn (ω)) ∩ SD = ∅. Therefore we have that P(∪m∈N Cm ) = 1 where \ Cm = {ω : ker (φn∗ (ω)) ∩ SD = ∅}. n≥m 68 Since (Cm )m∈N is an increasing sequence we have that lim P(Cm ) = P (∪m∈N Cm ) = 1 m→∞ But we have that θ −1 (Cm ) = Cm+1 , therefore P(Cm ) = P(C1 ) for all m ∈ N and thus P(C1 ) = 1. Therefore ker (φ1 ) ∩ SD = ∅ w.p.1 and thus ker(φ0 ) ∩ SD = ∅ with probability 1. 3.3.4 Limiting Results for Ergodic Processes with Invertible Dynamics Similar to the negative direction in the stationary case we have a version of lemma 3.2.10 for the negative kingman’s limit for the invertible ergodic case. Also version of theorem 3.2.14 holds for the invertible ergodic system. We state these results without proofs as proofs are similar to that of the mentioned lemma and theorem above. Corollary 3.3.9. Let (Φ (n) )n∈Z be as in 3.3.6. Assume that the original sequence (φn )n∈Z satis- fies the assumptions B’1 and B’2 (or just B’ due to lemma 3.3.8) then there exists a deterministic κ′ ∈ [0, 1) such that a.s 1 ∗ ln κ′ = lim ln c (φ−n ∗ ◦ . . . ◦ φ−1 ). (3.3.13) n→∞ n   Furthermore we have that c Φ −(n) → 0 almost surely as n → ∞. Corollary 3.3.10. Let (Φ (n) )n∈Z be an ergodic quantum process obtained from an invertible ergodic system as defined in 3.3.6 with assumptions B’1 and B’2 (or just B’ due to lemma 3.3.8). Then there are two-sided stationary sequences of (random) matrices (Zn )n∈Z and (Zn′′ )n∈Z such ′′ that Zn = Z1 ◦ θ n−1 , Zn′′ = Z−1 ◦ θ n+1 and 1. Z1 ∈ SoD almost surely and limn→∞ Ln converges (in trace norm) to Z1 almost surely. 2. For n ∈ Z we have that φn∗ · Zn+1 = Zn almost surely. 3. For any Y ∈ SD and k < n ∈ Z we have that (φk∗ ◦ . . . ◦ φn∗ ) · Y converges (as n → ∞) to Zk (in trace norm) almost surely.     4. For any Y ∈ SD and k < n in Z we have that d (φk∗ ◦ . . . ◦ φn∗ ) · Y , Zk ≤ c φk∗ ◦ . . . ◦ φn∗ . 5. Z−1′′ ∈ SoD almost surely and limn→−∞ Rn converges (in trace norm) to Z−1 ′′ almost surely. 69 6. For k ∈ Z, Zk′′ is the almost sure (trace) norm limit (as n → ∞) of (φk ◦ . . . ◦ φ−n ) · X for any X ∈ SD . 7. For k ∈ Z we have that Zk′′ = φk · Zk−1 ′′ almost surely. 8. For any Y ∈ SD and m < n in Z we have that d ((φn ◦ . . . ◦ φm ) · Y , Zn′′ ) ≤ c (φn ◦ . . . ◦ φm ). Recall that from corollary 3.3.4 we have that ln κ is (almost surely) constant. We shall (n) list some results that are consequences of ln κ being a constant. For m < n let Φm = φc ◦ . . . ◦ φm and let Pn,m for n, m ∈ Z be the rank one operator Pn,m (X) = tr [Zm X] Zn′′ where Zm and Zn′′ are as in theorem 3.3.10. Then we have the follwing result. The proof of the theorem below is similar to that of the 3.2.18. As such we omit the proof. We note that µ in the theorem can be taken as a deterministic function. Theorem 3.3.11. There is a sequence of rank one operators Pn,m such that for each x ∈ Z we have that 1 (n)   Φm − Pn,m ≤ Cµ,x µn−m almost surely (3.3.14) (n) tr Φm (I) for all m ≤ x and n ≥ x with Cµ,x finite almost surely and µ a constant in (0, 1). 3.4 IID Case and Invariant Measures In this section we study special case of the ergodic case. Given a sequence of IID (inde- pendent and identically distributed) positive operators φn we shall define the associated quantum process as φ(n) = φn ◦ . . . ◦ φ1 From lemma 2.2.7 it is reasonable to assume that the IID sequence is defined on an er- godic system. If in addition we also assume that the assumptions B1 and B2 hold we have that the same results for the ergodic case (such as corollary 3.3.5) hold true for the IID case. Therefore we wish not to reiterate the same results. However there are other interesting aspects of IID quantum processes. 70 Before we present any results in this section, we need the change of variable formula. This formula can be found in most standard probability theory books [21] [41]. Change of Variable Formula 3.4.1. Let X be a measurable map from a probability space (Ω, F , P) into some measurable space (S, S). Let µ denotes the law of x, i.e µ(A) = P◦X −1 (A) for any A ∈ S. Now for any measurable function f : (S, S) → (R, B(R)) with f ≥ 0 or E|f (X)| < ∞ one has Z Ef (X) = f (s) µ(ds) . S We have that as φ0∗ is a measurable map on (Ω, F , P) taking values in L(MD ) with the Borel σ -algebra B(L(MD )) induced by the operator norm and we have that φ0∗ induces a (probability) measure on L(MD ) via its push-forward (P ◦ (φ0∗ )−1 ). Furthermore the exis- tence of Z1 from 3.2.14 shows us that there is a measure on SD (with the Borel σ -algebra induced by the trace norm) which is the push-forward (P ◦ Z1−1 ). Since the elements in L(MD ) interacts with elements in SD through the projective action φ · X (for φ ∈ L(MD ) and X ∈ SD ) we may define the convolution of the two measures P ◦ φ0−1 and P ◦ Z1−1 as follows: Let µ = P ◦ (φ∗ )−1 and ν = P ◦ Z1−1 then for any bounded continuous real valued function f on SD we define Z Z ! µ ∗ ν(f ) = f (φ · X) dµ(φ) dν(X). Sd L(MD ) We may even define above for any measure ν on SD and we note that φ · X is well-defined in the sense that φ(X) = 0 happens in a µ−probability 0 event due to assumption B1. For a general probability measure ν on SD we have that Z Z ! µ ∗ ν(f ) = f (φ · X) dµ(φ) dν(M) (3.4.1) SD L(MD ) Z Z ! = f (φ0∗ (ω) · X) dP(ω) dν(M) from change of variable formula (3.4.2) Ω Z SD (E f (φ0∗ · M) ) dν(M)   = (3.4.3) SD 71 Since all φn have the same distribution we also have that for all n ∈ N ∪ {0} and any bounded continuous function f on SD Z µ ∗ ν(f ) = (E [f (φn∗ · M)]) dν(M) (3.4.4) SD A measure ν on SD is said to be µ-invariant if we have that, for all bounded Borel real-values functions f on SD one has Z µ ∗ ν(f ) = f (X) dν(X). (3.4.5) SD We now present the following theorem which states that the only µ-invariant proba- bility distribution on SD is the distributing of Z1 . Theorem 3.4.2. Let ν denote the distribution of Z1 then for any bounded Borel measurable real-valued function f on SD we have that Z µ ∗ ν(f ) = f (X)dν(X) SD furthermore for any µ-invariant probability measure v ′ on SD we have that ν ′ = ν. Proof. Let ν = P ◦ Z1−1 and µ = P ◦ (φ0∗ )−1 then consider the product probability space SD × L(MD ) with the product sigma algebra and the product probability measure ν ⊗ µ. Now consider the map X : ω 7→ (Z1 (ω), φ0∗ (ω)), we have that X is (B(SD ) ⊗ B(L(MD ), F )) = (B(SD × L(MD )), F )-measurable. Now for any Bounded Borel function f on SD define F(X, φ) = f (φ · X). Therefore we have that Z EF(X) = F(A, φ) d Law(X)(A, φ) SD ×L(MD ) Z Z Z = F(A, φ) dν ⊗ dµ(A, φ) = f (φ · A)dµ(φ)dν(A). (3.4.6) SD ×L(MD ) SD L(MD ) Where we have used the fact that φ0∗ and Z1 are independent (this is because Z1 ∈ σ (φk : k ≥ 1) and φ0 is independent of all φk for k ≥ 1) and therefore the joint distribution of φ0∗ and Z1 is the same as the product of the two distributions and the last equality is from 72 Fubini’s theorem. But on the other hand we have that Z Z Z EF(X) = f (φ0∗ (ω)) · Z1 )dP(ω) = f (Z0 ) dP(ω) = f (M) dν(M). (3.4.7) Ω Ω SD Here we have used that φ0∗ · Z1 = Z0 to get the second equality and the last equality is just d change of variables with the fact that Z1 = Z0 . So we have that 3.4.6 and 3.4.7 combined displays that the distribution of Z1 is invariant w.r.t. to the distribution of φ0∗ . Now let ν ′ be any µ invariant probability on SD and consider the following sequence of measurable maps for a fixed X ∈ SD . ω 7→ (ω, ω) 7→ (φ1∗ , φ2∗ ) 7→ (φ1∗ ◦ φ2∗ ) 7→ (φ1∗ ◦ φ2∗ ) · X 7→ f ((φ1∗ ◦ φ2∗ ) · X) | {z } | {z } L F Then we have that Z E[F ◦ L] = F(φ, ψ) dP ◦ L−1 (φ, ψ) (3.4.8) L(MD )×L(MD ) Z Z ! = F(φ, ψ) dP ◦ φ1∗ −1 (φ) dP ◦ φ2∗ −1 (ψ) (3.4.9) L(MD ) L(MD ) Z Z ! = f ((φ ◦ ψ) · X) dP ◦ φ1∗ −1 (φ) dP ◦ φ2∗ −1 (ψ) (3.4.10) L(MD ) L(MD ) Z Eω f ((φ1∗ (ω) ◦ ψ) · X) dP ◦ φ2∗ −1 (ψ).   = (3.4.11) L(MD ) In the above set of equalities we have used that (φi∗ )i∈N are all independent (so their joint distribution is the same as the product of distributions) with Fubini’s theorem. Now let g be the map on SD defined as g(M) = E[f (φ1∗ · M)]. We have that g is con- tinuous and bounded: Since f is bounded it is easy to see that g is also bounded. To see that g is continuous note that as f is continuous we have that for a sequence of matrices Mn such that Mn → M in SD we have that E[f (φ1∗ · Mn )] → E[f (φ1∗ · M)] by Dominated Convergence Theorem. Thus g is continuous. Since g is bounded and continuous we have that Z Z E f ((φ1∗ ◦ ψ) · X) dP ◦ φ2∗ −1 (ψ) dν ′ (X)   SD L(MD ) 73 Z Z = g(ψ · X) dP ◦ φ2∗ −1 (ψ) dν ′ (X) (3.4.12) SD L(MD ) Z Z = g(ψ · X) dP ◦ φ0∗ −1 (ψ) dν ′ (X) (3.4.13) SD L(MD ) Z = g(X) dν ′ (X) (3.4.14) SD Z = E[f (φ1∗ · M)] dν ′ (M) (3.4.15) ZSD Z = f (φ · M) dµ(φ) dν ′ (M) (3.4.16) S L(MD ) Z D = f (M) dν ′ (M). (3.4.17) SD Where we have used that the φi s have identical distributions in the second equality and that ν ′ is µ invariant in the third and the last equality. On the other hand we have that Z Z E f ((φ1∗ ◦ ψ) · X) dP ◦ φ2∗ −1 (ψ) dν ′ (X)   SD L(MD ) Z Z Z = f ((φ ◦ ψ) · X) dP ◦ (φ1∗ )−1 dP ◦ φ2∗ −1 (ψ) dν ′ (X) (3.4.18) SD L(MD ) L(MD ) Z Z = f ((φ ◦ ψ) · X) dP ◦ (φ1∗ , φ2∗ )−1 dν ′ (X) (3.4.19) SD L(MD )×L(MD ) Z = E[(φ1∗ ◦ φ2∗ ) · X] dν ′ (X). (3.4.20) SD Here the second equality uses Fubini’s theorem together with the fact that the joint distri- bution of (φ1∗ , φ2∗ ) is the same as the product of their distributions due to independence. Therefore we have from 3.4.15, 3.4.17 and 3.4.20 that Z Z ∗ ∗ ′ E[f ((φ1 ◦ φ2 ) · M)] dν (M) = E[f (φ1∗ · M)] dν ′ (M). SD SD Proceeding in this manner we see that Z Z Z (n)∗ ′ E[f (φ · M)] dν (M) = E[f (φ1∗ · M)] ′ dν (M) = f (M) dν ′ (M). SD SD SD Now we take the limit n → ∞. Then by dominated convergence theorem and from corol- lary 3.3.5 we have that Z Z ′ E[f (Z1 )] dν (M) = E[f (Z1 )] = f (M) dν ′ (M) SD SD 74 Since the above equality holds for all bounded continuous functions we have that the two measures P ◦ Z1−1 and ν ′ are equal. Now if we further assume that the process is built from a two sided IID sequence we have that the results in corollary 3.3.10 hold for the IID case. Then similar to the proof of ′′ theorem 3.4.2 we have the following theorem for the distribution of Z−1 and φ1 , proof of which is essentially the same as the proof of theorem 3.4.2 as such we state the result and omit the proof. ′′ Theorem 3.4.3. Let η denote the distribution of Z−1 and let υ denote the distribution of φ0 then for any bounded Borel measurable real-valued function f on SD we have that Z υ ∗ η(f ) = f (X)dη(X) SD furthermore for any υ-invariant probability measure η ′ on SD we have that η ′ = η. 75 CHAPTER 4 CONTINUOUS PARAMETER QUANTUM PROCESSES 4.1 Notations and Assumptions In this chapter we will study the notion of a quantum process in the continuous time parameter. We shall assume that the underlying probability space is a standard Borel space. A standard Borel space is the Borel space associated to a separable completely metrizable topological space i.e. a Polish space. We refer the reader to [53] some results on standard Borel Spaces. We assume that on this probability space we have a one parameter group of measure preserving transformations (θg )g∈R such that; 1. Fora all g1 , g2 ∈ R, θg1 ◦ θg2 = θg1 +g2 . 2. For the identity element e ∈ R, θe (ω) = ω for all ω ∈ Ω. We see that for a such a family of measure preserving transformations each θg must be bijective. To see that, let g ∈ R and and consider the map θg we must have that θg ◦ θ−g = θ−g ◦ θg = θ0 where θ0 is the identity transformation therefore θg is invertible with the inverse θ−g and for any ω ∈ Ω where Ω denotes the underlying probability space we have that θg (θ−g (ω)) = ω so θg is also surjective. Since the underlying probability space is assumed to be a standard Borel space and R is a locally compact separable space we have due to a theorem by Mackey in [54] the two definition; w-ergodic and s-ergodic (see 2.1.7, 2.1.8) are equivalent. Since both definitions are equivalent we shall simply call the family of measure preserving transformations θg )g∈R on a standard Borel space as ergodic if the family is either w-ergodic or s-ergodic. Now we define a quantum process in continuous time parameter: Definition 4.1.1. A continuous parameter quantum process is a family of positive map val- ued random variables (φs,t )(s,t)∈R×R defined on a standard Borel space (Ω, F , P, (θt )t∈R ) where (θt )t∈R is a one parameter group of preserving transformations on Ω such that P0. For r < s < t we have that φs,t (ω) ◦ φr,s (ω) = φr,t (ω) for all ω ∈ Ω. 76 P1. φs+h,t+h (ω) = φs,t (θh (ω)) for all ω ∈ Ω. Similar to the discrete we shall assume the following Assumption Q1. There exists an event E0 with probability one such that on E0 we have that for all s, t ∈ R, ker(φs t) ∩ SD = ker(φs t ∗ ) ∩ SD = ∅. Furthermore similar to discrete case we assume that the (sub)process (φ0,t )t>0 becomes strictly positive in finite time almost surely. We shall see that this together with assump- tion Q1 will make the sub-processes eventually strictly positive w.p. 1. Assumption Q2. For τ(ω) = inf{φ0,t is strictly positive} we have τ < ∞ almost surely. t>0 With properties Q1 and Q2 we have the existence of an event E with probability one, on which both properties Q1 and Q2 are satisfied. Lemma 4.1.2. On E we have that for all δ > 0, φ0,τ+δ is strictly positive. Proof. Let ω ∈ E. Then from the definition of τ(ω) we have that there exists 0 < t0 < τ(ω) + δ such that φ0,t0 (ω) is strictly positive. Now we have φ0,τ(ω)+δ (ω) = φt0 ,τ(ω)+δ (ω) ◦ φ0,t0 (ω) (4.1.1) Since φ0,t0 (ω) is strictly positive we have that from proposition 2.7.2 that φt0 ,τ(ω)+δ (ω) ◦ φ0,t0 (ω) is also strictly positive. Lemma 4.1.2 tells us that under the given assumptions we get that φ0,t becomes strictly positive a.s. in finite time and stays strictly positive thereafter. We now show that the same will hold for the sub-process started at an arbitrary time s ∈ R; (φs,t )s 0 : φ0,t (θs (ω)) is strictly positive.} (4.1.2) = inf{t > s : φs,t (ω) is strictly positive.} − s. (4.1.3) This yields us that under assumption Q2, τs defined below is a.s. finite. τs (ω) := inf{φs,t+δ is strictly positive ∀δ > 0.}. (4.1.4) t>s We also get that τs (ω) = τ(θs ) + s a.s. Similar to the proof of lemma 4.1.2 we also have that for δ > 0, φs,τs +δ is strictly positive. 1  Lemma 4.1.4. Let ft (ω) = ln c φ0,t (ω) . Then we have that there exists κ : Ω → [0, 1] such t that a.s 1  ln κ = lim ln c φ0,t (ω) (4.1.5) t→∞ t and that κ < 1 almost surely.  Proof. Let cm,n := ln c φm,n for m < n ∈ N. Then the discrete parameter process (cs,t )s N we have that c φo,t < eMt . Since this holds for any t > N we can take limit   t → ∞ and since M < 0 we see that limt→∞ c φ0,t (ω) = 0. Now assume that κ(ω) ∈ (0, 1) then we have that ln κ(ω) is finite and negative. Now for −(1/2) ln κ(ω) > 0 we have that  ln c φ0,t (ω) < t[ln κ(ω) − (1/2) ln κ(ω)] eventually for large t. Thus for t large enough we   have that c φ0,t (ω) ≤ exp{{(t/2) ln κ(ω)}}. Taking limit t → ∞ we see that c φ0,t (ω) → 0.  Corollary 4.1.6. For any s ∈ R we have that limt→∞ c φs,t → 0 with probability 1. Proof. For t > s we have that φs,t = φs,s+(t−s) = φ0,t−s ◦ θs . Now as θs is measure preserving   we have that 1 = P{ω : limδ→∞ c φ0,δ (ω) = 0} = P{ω : limt→∞ c φ0,t (θs (ω)) = 0} where δ = t − s. Similar to the discrete case under the stronger condition that (θt )t∈R is an ergodic family, we shall show that the limit ξ = ln κ (and hence κ) is almost surely constant with the constant being (strictly) less than 0. To this end we require the following lemma. Lemma 4.1.7. Let (φs,t )(s,t)∈R×R be a family of positive map valued random variables satisfying conditions P0, P1 as defined in 4.1.1. Furthermore assume that the family satisfy the conditions Q1 and Q2. Then there exists some N0 ∈ N such that P{ω : φ0,N0 (ω) is strictly positive} > 0. Proof. Since τ is finite a.s. we have that [  X 1 = P{τ < ∞} = P {ω : τ(ω) < n} ≤ P{ω : τ(ω) < n}. (4.1.11) n∈N n∈N So it must be the case that there is some N0 ∈ N such that P{ω : τ(ω) < N0 } > 0. Now by lemma 4.1.2 we have that φo,N0 (ω) is also strictly positive whenever ω ∈ {τ < N0 }. With the previous lemma we have the following corollary to lemma 4.1.4 under the stronger condition that the one parameter group of measure preserving transformations are ergodic. 80 Corollary 4.1.8. If the family of measure preserving transformations (θt )t∈R are ergodic then we have that κ is almost surely a non-negative constant less than 1. Proof. Let ξ = ln κ where ξ and κ are as in the proof of lemma 4.1.4. Since E[c0,m+n ] ≤ E[c0,n ] + E[cn,n+m ] = E[c0,n ] + E[c0,m ] we also have from theory of subadditive functions ([35]) that 1 1 lim E[c0,n ] = inf E[c0,n ], (4.1.12) n→∞n n∈N n n∈N 1 and from [45] we have that for ξ = lim c0,n n→∞n n∈N 1 E[ξ] = lim E[c0,n ] = inf E[c0,n ]. (4.1.13) n→∞ n∈N n n∈N Let δ > 0 then observe that:    ln c φ0,t+δ (ω) t ln c φ0,δ (ω) ln c φ0,t (θδ (ω))   ≤ + . (4.1.14) t+δ t+δ t t Now taking limits we see that ξ ≤ ξ ◦ θδ almost surely. Therefore we have that P{ω : ξ(ω) , ξ ◦ θδ (ω)} = P{ω : ξ(ω) < κ ◦ θδ (ω)} (4.1.15) [n o =P ω : ξ(ω) ≤ r < (ξ ◦ θδ )(ω) (4.1.16) r∈Q X   ≤ P {ω : (ξ ◦ θδ )(ω) > r} \ {ω : ξ(ω) > r} = 0 . (4.1.17) r∈Q The last equality comes from the fact that P(A \ B) = P(A) − P(A ∩ B) with A = {ω : (ξ ◦ θδ )(ω) > r} and B = {ω : ξ(ω) > r}. Now since ξ ≤ ξ ◦ θδ a.s we have that B ⊂ A a.s. Hence P(A ∩ B) = P(B) and since θδ is measure preserving we have that P(A) = P(B). This shows us that for any δ > 0 we have that ξ = ξ ◦ θδ almost surely. Now for γ < 0 let δ = −γ and for sufficiently large t    ln c φ−δ,t−δ ln c φ−δ,0 t − δ ln c φ0,t−δ ≤ + · . (4.1.18) t t t t−δ Taking limit t → ∞ we have that ξ ◦ γ ≤ ξ almost surely. Similar to the part above we can prove that for γ < 0, ξ ◦ θγ = ξ almost surely. This means that ξ is essentially θδ invariant 81 for all δ ∈ R and thus ξ is a.s. a constant. Now since ln κ = ξ we have that κ too is almost surely a constant. Finally from lemma 4.1.7 we see that there exists N0 ∈ N such that for φ0,N0 is strictly   positive with positive probability. This means that E[ln c φ0,N0 ] < 0. Therefore by 4.1.12 we have that E[ξ] and hence E[ln κ] is strictly smaller than 0. But since κ is a.s. constant a.s we have that ln κ = E[ln κ] < 0 and thus 0 ≤ κ < 1. 4.2 Limiting Results in Forward Time In the regime that τ(ω) is finite we have that φ0,t (ω) becomes strictly positive in finite time t. Since any strictly positive map is irreducible in the sense in [23] we have that left and right Perron-Frobenius eigenmatrices, Rt and Lt , exists for sufficiently large t. ∗ φ0,t (ω)(Rt (ω)) = Λt (ω)Rt (ω) and φ0,t (Lt (ω)) = Λt Lt (ω). (4.2.1) ∗ Here Λn (ω) denotes the spectral radius of φ0,t (ω) (equiv. of φ0,t (ω)). We will omit ω from these notations for ease. We further normalize Ln and Rn so that they are of trace 1. So Rn and Ln in equation 4.2.1 are uniquely determined by the relations in 4.2.1 and belonging to S0D . Theorem 4.2.1. Let (φs,t )(s,t)∈R×R be a family of positive map valued random variables as ∗ defined in 4.1.1 with the assumptions Q1 and Q2. Let φ0,t and Lt be as in equation 4.2.1. Then as t → ∞, Lt converges a.s. to a limit Z0 in d−metric (see 2.6.2) and also in trace norm ∥ ∥. We have that Z0 ∈ S0D almost surely. Furthermore for Zs := Z0 ◦ θs we have that;   ∗  ∗ 1. ∀Y ∈ SD , we have that d φ0,t · Y , Z0 ≤ c φ0,t and thus limt→∞ φ0,t · Y = Z0 a.s. ∗ 2. For any s ∈ R and Y ∈ SD , φs,t · Y converges to Zs almost surely as t → ∞. ∗ 3. For any s ∈ R and t > s we have that φs,t · Zt = Zs .   4. ∀Y ∈ SoD , we have that d φs,t∗  · Y , Zs ≤ c φs,t .   ∗ ∗ ∗ ∗ Proof. Let Bt = φ0,t · SD . Then we have that for h > 0, Bt+h = φ0,t+h · SD = φ0,t ◦ φt,t+h · SD . So Bt+h ⊆ Bt . Therefore we have that the collection (Bt (ω))t>τ0 (ω) is a sequence of d- 82 compact subsets in SD with the finite intersection property. Where the compactness follows from lemma 2.6.2 part 6 as φ0,t (ω) is strictly positive for sufficiently large t we have that Bt (ω) ⊂ SoD for large t (Since Bt is compact in norm topology we have Bt is compact in d-metric topology for large t). Therefore we have that ∩t>τ(ω) Bt (ω) is non-empty a.s. Furthermore we have that diam(Bt (ω)) ≤ c(φ0,t (ω)). Therefore by corollary 4.1.5 we must have that ∩t>τ(ω) Bt (ω) = {Z0 (ω)} for some Z0 (ω) ∈ SD . This gives us the existence of Z0 a.s. Furthermore since φ0,t (ω) is strictly positive for sufficiently large t we have that Bt ⊂ SoD resulting in Z0 ∈ SoD almost surely. Due to 4.2.1 we also have that Lt (ω) ∈ Bt (ω) for t sufficiently large. This gives us that d (Lt , Z) ≤ diam(Bt ) ≤ c(φ0,t ). Hence Lt → Z0 almost surely as t → ∞. Now we proceed to prove the other properties.     ∗ ∗ For Y ∈ SD we have that d φ0,t · Y , Z0 ≤ diam(Bt ) hence we have that d φ0,t · Y , Z0 → ∗ 0 as t → ∞. This together with lemma 2.6.2 we get the desired convergence limt→∞ φ0,t · Y = Z0 a.s. for all Y ∈ SD . ∗ Let s ∈ R then let Zs = Z0 ◦ θs . The as φ0,t · Y converges to Z0 in d-metric (and thus in trace norm) almost surely we have, as θs is measure preserving, it must be the case that ∗ φs,t+s · Y converges to Zs almost surely for any Y ∈ SD . Therefore we have that     ∗ ∗ ∗ ∗ ∗ ∗ Zs = lim φs,t · Y = lim (φs,t ′ ◦ φt′ ,t ) · Y = φs,t′ · lim φt′ ,t · Y = φs,t ′ · Zt ′ . t→∞ t→∞ t→∞ Finally we have that for s < t in R and Y ∈ SD       ∗ ∗ ∗ ∗  d φS,t · Y , Zs = d φs,t · Y , φs,t · Zt ≤ c φs,t = c φs,t where the last equality comes from lemma 2.8.1. This concludes the proof. 4.3 Limiting Results in Backward Time Similar to the discrete case we shall assume that the sub-processes φt,0 for t < 0 be- comes strictly positive as for finite t < 0 and thus similar to the forward time the sub- processes will stay strictly positive there after. Therefore we introduce a new assumption for the backward time case: 83 Q3. For τ ′ (ω) := inf{s > 0 : φ−s,0 is strictly positive.} we have that τ ′ is finite a.s. Similar to the forward time case assumption Q3 will guarantee that the sub-processes φ−s,t for s > 0 shall become eventually strictly positive as s → ∞ with probability one. With this assumption we have the existence of left and right Perron-Frobenius eigenma- trices, Rt and Lt , for sufficiently large |t| almost surely. ∗ φ0,t (ω)(Rt (ω)) = Λt (ω)Rt (ω) and φ0,t (Lt (ω)) = Λt Lt (ω) for t > 0 (4.3.1) ∗ φt,0 (ω)(Rt (ω)) = Λt (ω)Rt (ω) and φt,0 (Lt (ω)) = Λt Lt (ω) for t < 0 (4.3.2) We normalize Lt and Rt so that they are of norm 1. By redefining a process ψs,t for s < t as ∗ ψs,t = φ−t,−s (4.3.3) ∗ we have that ψs,t is strictly positive if and only if ψs,t is strictly positive and hence φ−t,−s is strictly positive. Furthermore we have that the process (ψs,t )s0 defined in 4.3.3 with the assumptions Q3 and Q1 then we have that the following limit exists with probability 1. a.s. 1 1  ∗  ln κ′ = lim  ln c ψ0,s = lim ln c φt,0 . (4.3.4) s→∞ s t→−∞ t   ∗ Furthermore κ′ ∈ [0, 1) almost surely. We also have that limt→−∞ c (φt,0 ) = 0 almost surely   ∗ and thus for any s ∈ R, limn→−∞ c (φt,s ) = 0 almost surely. If in addition the one parameter 84 group of measure preserving transformations (θt )t∈R are ergodic we have that κ′ is almost surely a constant with κ′ < 1 almost surely. Theorem 4.3.2. Let Rt be as in equation 4.2.1. Then as t → −∞, Rt converges a.s. to a limit Z0′ , such that Z0′ ∈ SoD . Furthermore for Zs′ := Z0′ ◦ θ−s we have that;   1. ∀Y ∈ SD , we have that d φt,0 · Y , Z0′ ≤ c φt,0 whenever t < 0 and thus limt→−∞ φt,0 ·  Y = Z0′ a.s. 2. For any s ∈ R and Y ∈ SD we have that φt,−s ·Y converges to Zs′ almost surely as t → −∞. ′ 3. For any t < −s we have that Zs′ = φt,−s · Z−t . Remark 4.3.3. Similar to the discrete case it is convenient to re-index the random matrices Zs′ so that the re-indexed matrices, denoted by Zs′′ = Z−s ′ . So the properties listed in the theorem above will read:   1. ∀Y ∈ SD , we have that d φt,0 · Y , Z0′ ≤ c φt,0 and thus limt→−∞ φt,0 ·Y = Z0′′ a.s. where  Z0′′ is the limit of the eigen-matrices Rt as t → −∞. 2. For any s ∈ R and Y ∈ SD we have that φt,s · Y converges to Zs′′ almost surely as t → −∞. 3. For any t < s we have that Zs′′ = φt,s · Zt′′ . 4.4 Other Asymptotic Behaviour With the existence of these random matrices Zt and Zt′′ we obtain the following theo- rem which states that when t − s is large the operator φs,t is well approximated by a rank one operator. We note that this is the continuous extension of theorem 2 of [55] for the stationary case. First we need the following lemma, proof of which is similar to that of the proof of lemma 3.2.16 and hence omitted. Lemma 4.4.1. For s < t and X ∈ SD we have that   φs,t (X) ′  tr[φs,t (X)]  − tr[Zs X]Z−t ≤ 2c(φs,t )1 +  ≤ 4c(φs,t ) (4.4.1)  ∗ ∗ tr[φs,t (I)] tr[φs,t (I)]  almost surely. 85 With the help of the lemma above we shall now present the following theorem. We need some new notations. For s < t let Ps,t denoted the rank-one operator Ps,t (X) = tr[Zs X]Zt′′ (4.4.2) where Zs and Zt′′ are as in lemma 4.2.1 and remark 4.3.3. We have that for t − s large, the operator φs,t is well approximated by Ps,t . We also define N1 and N2 on the sub processes 1 ′ (φ0,n )n∈N and (φ−n,0 )n∈N as follows. First let µ(ω) = 2 (max{κ(ω), κ (ω)} + 1). Then µ is measurable as κ and κ′ are measurable and µ(ω) ∈ [0, 1) almost surely. Now let N1 (ω) = inf{n ∈ N : c φ0,n+k (ω) − (µ(ω))n+k < 0 for all k ∈ N ∪ {0}}  (4.4.3) and   ∗ N2 (ω) = inf{n ∈ N : c φ−n−k,0 (ω) − (µ(ω))n+k < 0 for all k ∈ N ∪ {0}}. (4.4.4) Then we must have that both N1 and N2 are measurable (similar to the discrete case)  and finite almost surely. This is because as (1/n) ln c φ0,n → ln κ almost surely we must  have that almost surely for ω there is some Nω such that (1/n) ln c φ0,n (ω) < ln µ(ω) and thus c φ0,n (ω) < µ(ω)n for all n ∈ N with n ≥ Nω . Therefore we have that N1 ≤ Nω almost   surely. Similarly we can see that N2 is finite almost surely: we have that (1/n) ln c ψ0,n → ln κ′ almost surely. Thus almost surely for ω there is some N ′ (ω) such that c ψ0,n < µ(ω)n  for all n ≥ N ′ (ω). Therefore we have that N2 ≤ N ′ almost surely. Theorem 4.4.2. There exists a continuous parameter process of rank one operators Ps,t , mea- surable maps µ : ω → (0, 1) and D : Ω → R such that for any x ∈ R we have that φs,t ∗ − Ps,t ≤ D(θx )(µ(θx ))t−s almost surely (4.4.5) tr φs,t (I) for all s ≤ x ≤ t with D is finite almost surely. Furthermore if the one parameter group of measure preserving transformations are assumed to be ergodic we have that µ is almost surely constant. 86 Proof. Similar to the proof of theorem 3.2.18 we note that to obtain 4.4.5, it is enough to establish φs,t (X) ∗ − Ps,t (X) ≤ Cµ.x µt−s ∥X∥ (4.4.6) tr φs,t (I) for any X ∈ SD with Cµ,x is an almost surely finite measurable map. Now let N1 , N2 and  µ be as in the discussion above. Then we have that almost surely for ω ∈ Ω, c φ0,n (ω) ≤ (µ(ω))n for all n ≥ N1 (ω). But since µ(ω) < 1 a.s. we have that (µ(ω))N1 (ω) ≤ 1 and therefore c φ0,n (ω) ≤ (µ(ω))n ≤ (µ(ω))n (µ(ω))−N1 (ω)  for all n ≥ N1 (ω). Now for n ≤ N1 we have as µ(ω) < 1 almost surely, (µ(ω))N1 (ω) ≤ (µ(ω))n and thus c φo,n (ω) ≤ 1 ≤ (µ(ω))n (µ(ω))−N1 (ω)  for all 0 ≤ n ≤ N1 (ω). This means that for all n ∈ N ∪ {0} we have that c φ0,n (ω) ≤ (µ(ω))n (µ(ω))−N1 (ω)  almost surely. This gives us the existence of an event of probability 1 on which the above inequality holds for all n ∈ N ∪ {0} everywhere. Now let ω be in that event, then since for t1 ∈ R with           t1 ≥ 0 we have that c φ0,t1 ≤ c φ0,⌊t1 ⌋ c φ⌊t1 ⌋,t1 ≤ c φ0,⌊t1 ⌋ , we have that c φ0,t1 (ω) ≤ µ(ω))⌊t1 ⌋ (µ(ω))−N1 (ω) almost surely.   Now let s ≤ x ≤ t then we have t −x ≥ 0. Since we have that for any t1 ≥ 0, c φ0,t1 (ω) ≤ µ(ω))⌊t1 ⌋ (µ(ω))−N1 (ω) w.p. 1 it must be the case that c φ0,t−x (ω) ≤ µ(ω))⌊t−x⌋ (µ(ω))−N1 (ω)  w.p. 1. Now as θx is measure preserving we see by translating that c φx,t (ω) ≤ µ(θx (ω)))⌊t−x⌋ (µ(θx (ω)))−N1 (θx (ω)) with probability 1.  (4.4.7) Now from the previous discussion we also have that for almost every ω ∈ Ω there is   ∗ N2 (ω) ∈ N such that c φ−n,0 < (µ(ω))n for all n ≥ N2 (ω). Similar to the previous part this would imply the existence of an event of probability 1 such that for any ω ∈ E we have that c φ−n,0 ≤ (µ(ω))n (µ(ω))−N2 (ω)  for all n ∈ N ∪ {0} 87         Therefore for any s1 ∈ R with s1 ≤ 0 we have that c φs1 ,0 ≤ c φ⌈s1 ⌉,0 c φs1 ,⌈s1 ⌉ ≤ c φ⌈s1 ⌉,0 .   Therefore for any s1 ∈ R with s1 ≤ 0 we must have that c φs1 ,0 ≤ (µ(ω))−⌈s1 ⌉ (µ(ω))−N2 (ω)  almost surely. Now as s ≤ x we have that s − x ≤ 0 and therefore we have that c φs−x,0 ≤ (µ(ω))−⌈s−x⌉ (µ(ω))−N2 (ω) almost surely, Now as θx is measure preserving we have by trans- lating that c φs,x ≤ (µ(θx (ω)))−⌈s−x⌉ (µ(θx (ω)))−N2 (θx (ω))  w.p.1. (4.4.8) Combining 4.4.7 and 4.4.8 we see that for s ≤ x ≤ t we have    c φs,t ≤ c φs,x c φx,t (4.4.9) ≤ µ(θx (ω)))⌊t−x⌋ (µ(θx (ω)))−N1 (θx (ω)) (µ(θx (ω)))−⌈s−x⌉ (µ(θx (ω)))−N2 (θx (ω)) (4.4.10) ≤ C(θx (ω))(µ(θx (ω)))⌊t−x⌋−⌈s−x⌉ (4.4.11) ≤ C(θx (ω))(µ(θx (ω)))t−s−2 (4.4.12) ≤ C(θx (ω))(µ(θx (ω)))−2 (µ(θx (ω)))t−s . (4.4.13) Here the last equality uses that t−x ≤ ⌊t−x⌋+1 and s−x ≥ ⌈s−x⌉−1. Therefore ⌊t−x⌋−⌈s−x⌉ ≥ t − s − 2. But as µ < 1 almost surely we have that µ⌊t−x⌋−⌈s−x⌉ ≤ µt−s−2 almost surely. In the above set of inequalities we have denoting by C(ω) = (µ(ω))−(N1 (ω)+N2 (ω)) . This combined with lemma 4.4.1 yields the desired results as explained in the beginning of the proof. 88 CHAPTER 5 LLN FOR ERGODIC QUANTUM PROCESSES 5.1 Linear Cocyles and Lyapunov Exponent In this section we introduce the notion of a linear cocycle that will be useful in both the proposed law of large numbers theorem and the central limit theorem. We shall define a linear cocycle in the context of the thesis. Definition 5.1.1 (Linear Cocycle). Let (Ω, F , P) be a probability space with a measure pre- serving transformation θ Let φ : Ω → L(MD ) be a measurable map. A linear cocycle defined by φ over θ is the transformation F : Ω × MD → Ω × MD (5.1.1) (ω, X) 7→ (θ(ω), φ(ω)X). Clearly for n ≥ 1 we have that F n (ω, X) = (θ n (ω), φ(n) (ω)X) where φ(n) (ω)X = (φ(θ n−1 (ω)) ◦ . . . ◦ φ(ω))X. A cocycle is said to be P-integrable if E([ln φ(ω) )+ ] < ∞. For an integrable cocycle we have, since norms are sub-multiplicative, by Kingman’s subadditive ergodic theorem [45] the limit 1 λ(ω) := lim ln φ(n) (ω) n→∞ n exists almost surely. This limit shall be called the top (maximal) Lyapunov exponent of the cocycle. 5.2 Law of Large Numbers In this chapter we will derive a Law of Large Numbers (LLN) for ergodic quantum processes as defined in 3.3.1 that tells us under certain integrability assumptions, the asymptomatic behaviour of ln⟨Y , Φ (n) X⟩ for X, Y ∈ SD is governed by the top Lyapunov 89 exponent. To obtain a meaningful Law of Large Numbers we need some integrability assumptions. First recall that for a map φ ∈ L(MD ) we define v(φ) as the infimum over all X ∈ SD of φ(X) (see 2.4.9). With this notation we assume the integrability of the following two random variables. Assumption I. We have E[| ln ∥φ0∗ ∥|] < ∞ and E[| ln v(φ0∗ )|] < ∞. We shall now state the proposed Law of Large Numbers (LLN) for an ergodic quantum process. LLN. Theorem 5.2.1. Let (φn )n∈N be a sequence of ergodic positive maps and Φ (n) = φn ◦. . . φ1 as defined in 3.3.1. Assume that the sequence (φn )n∈N satisfy the assumptions A1, A2 and I then we have that 1 lim sup ln⟨Y , Φ (n) (X)⟩ − l = 0 a.s. (5.2.1) n→∞ X,Y ∈S D n Where l is the top (maximal) Lyapunov exponent of the cocycle define by φ0 over θ. Further- more l is almost surely constant and we have 1 1 lim ln Φ (n) = lim ln Λn = l = E ln φ0∗ (Z1 ) . (5.2.2) n→∞ n n→∞ n We recall that Λn is the spectral radius of Φ (n) and Z1 is the limit of the left Per- ron–Frobenius eigen-matrices Ln as n → ∞ and Zn = Z1 ◦ θ n−1 . Remark 5.2.2. We take ln⟨Y , Φ (n) (X)⟩ = −∞ if ⟨Y , Φ (n) (X)⟩ = 0; by Assumption B1 and B2 this happens for at most finitely many n (see lemma 3.3.3). By Assumption I, l = E[ln ∥φ0∗ (Z1 )∥] is finite. To prove this result we need the following new notations. Let Φ (n) be the random map constructed by an ergodic sequence of positive maps as in 3.3.1. Under assumptions B1 and B2 (or equivalently under the assumptions B’1 and B’2 in the invertible case) recall that τ, the smallest time the sequence Φ (n) becomes strictly positive (and stays strictly 90 positive there after), is finite almost surely. With this information in mind we define the following two random sequences n o Dn (ω) = sup ln⟨Y , (Φ (n) (ω))(X)⟩ − ln (Φ (n)∗ (ω))(Y ) , (5.2.3) X,Y ∈SD and  Xn  (n) ∗ En (ω) = sup ln⟨Y , (Φω (ω))(X)⟩ − ln (φk,ω (ω))(Zk+1 (ω)) . (5.2.4) X,Y ∈SD k=1 5.3 Limits for Fractional Contraction Coefficient Before we provide a proof of the Law of Large Numbers Theorem we present the fol- lowing lemma that considers the contraction obtained from only a fraction of the process. This is described in the following Lemma 5.3.1. Let (ϕn )n≥1 and Φ n be as in eq. 3.3.1. Let α ∈ (0, 1) and let nα = ⌊(1 − α)n⌋, the integer part of (1 − α)n. If assumption B1 and B2 holds, then 1 lim ln c(ϕn ◦ · · · ◦ ϕnα +1 ) = α ln κ almost surely, (5.3.1) n→∞ n where κ is the deterministic constant in corollary 3.3.4. Proof. First note that, by proposition 2.8.1, we have     ln c ϕn ◦ . . . ◦ ϕnα +1 ≥ ln c (ϕn ◦ . . . ◦ ϕ1 ) − ln c ϕnα ◦ . . . ◦ ϕ1 . (5.3.2) Thus, by corollary 3.3.4, 1   lim inf ln c ϕn ◦ . . . ◦ ϕnα +1 ≥ α ln κ almost surely. (5.3.3) n→∞ n To prove the complementary upper bound, i.e., that 1   lim sup ln c ϕn ◦ . . . ◦ ϕnα +1 ≤ α ln κ , (5.3.4) n→∞ n we will show that for each m ∈ N 1   1   lim sup ln c ϕn ◦ . . . ◦ ϕnα +1 ≤ α E[ln c Φ (m) ] almost surely. (5.3.5) n→∞ n m 91 Eq. (5.3.4) will then follow by proposition 2.8.1 and theorem 3.2.4. Let m ∈ N be fixed and consider n ∈ N large enough that n−nα > 2m. Let p(n) = ⌊ nαm+m ⌋ and let q = q(n) ∈ N and r = r(n) ∈ {0, 1, . . . , m − 1} be defined by n = qm + r. Then, nα + 1 ≤ p(n)m + 1 ≤ nα + m < n − m + 1 ≤ q(n)m . (5.3.6) Since ln c (ϕ) ≤ 0 for any ϕ ∈ L(MD ), we have, using proposition 2.8.1, that     ln c ϕn ◦ . . . ◦ ϕnα +1 ≤ ln c ϕq(n)m+j ◦ . . . ◦ ϕp(n)m+j+1 (5.3.7) for any 0 ≤ j ≤ m−1, where eq. (5.3.6) guarantees that p(n)m+j+1 ≥ 1 and the composition on the right hand side has non-zero number of factors. Using, proposition 2.8.1 again, we find that q(n)−1 X q(n)−1 X     ln c ϕn ◦ . . . ◦ ϕnα +1 ≤ ln c ϕkm+j+m ◦ . . . ◦ ϕkm+j+1 = ln c (ϕm ◦ . . . ◦ ϕ1 )◦θ km+j . k=p(n) k=p(n) Since this holds for any j ∈ {0, 1, . . . , m − 1}, we have m−1 q(n)−1   1X X ln c ϕn ◦ . . . ◦ ϕnα +1 ≤ ln c (ϕm ◦ . . . ◦ ϕ1 ) ◦ θ km+j m j=0 k=p(n) q(n)m−1 1 X = ln c (ϕm ◦ . . . ◦ ϕ1 ) ◦ θ i m i=p(n)m q(n)m−1 p(n)m−1 X 1   X 1   = ln c Φ (m) ◦ θ i − ln c Φ (m) ◦ θ i . m m i=0 i=0  + 1 Since m ln c (ϕ m ◦ . . . ◦ ϕ 1 ) ∈ L1 (Ω) (where (·)+ denotes the positive part), eq. (5.3.4) fol- lows from the Birkoff ergodic theorem. 5.4 Proof of the Law of Large Numbers Theorem We see that the proof of Law of Large Numbers theorem will follow if we prove that 1 1. limn→∞ En = 0 n a.s 2. limn→∞ n1 nk=1 ln φk∗ (Zk+1 ) = E ln φ0∗ (Z1 ) := l ′ . P 92 3. l = l ′ . With the notations above we see that by assumption I we have E[ln φk∗ (Zk+1 ) ] < ∞ for each k ∈ N. Thus by Birkhoff’s ergodic theorem we have n 1X a.s lim ln φk∗ (Zk+1 ) = E ln φ0∗ (Z1 ) := l ′ . n→∞ n k=1 Now if we prove that limn→∞ n1 En = 0 then we have that 1 lim sup ln⟨Y , Φ (n) (X)⟩ − l ′ = 0 a.s. (5.4.1) n→∞ X,Y ∈S D n Now let X ∈ SD and let Y = I/D. Then from 5.4.1 we have that 1 I (n) 1 lim ln⟨ , Φ (X)⟩ = lim ln Φ (n) (X) = l ′ a. s. (5.4.2) n→∞ n D n→∞ n But φ(X) ≤ φ whenever ∥X∥ = 1 so we have that l ′ ≤ l. But as Λn = ⟨Ln , Φ (n) (I)⟩ we have that (1/D)Λn = ⟨Ln , Φ (n) (I/D)⟩. Then, again by 5.4.1 we have that 1 1 1 lim ln⟨Ln , Φ (n) (I/D)⟩ = lim ln⟨Ln , Φ (n) (I)⟩ = lim ln Λn = l ′ (5.4.3) n→∞ n n→∞ n n→∞ n Since norms are sub-multiplicative we have that fn := ln Φ (n) are sub-additive in the sense that fn+m ≤ fn + fm ◦ θ n for all m, n ∈ N. Therefore we have that the limit 1 lim ln φ(n) = l n→∞ n exists by Kingman’s sub-additive ergodic theorem. But we have that Φ (n) ≤ D Φ (n) ∞ where ∥ ∥∞ is the norm on L(MD ) generated by the ∥ ∥∞ on MD . Since Φ (n) ∞ = Λn we have that l ≤ l ′ from 5.4.3. Remark 5.4.1. Theorem 5.2.1 is closely related in spirit to the Furstenberg-Kesten theorem [28] and Oseledet’s Theorem [60] (see also [29]). By the Furstenberg-Kesten Theorem, the following limit exsists 1 lim ln Φ (n) a.s. = l a.s., n→∞ n 93 where l is a deterministic quantity called the top Lyapunov exponent. By Oseledet’s Theorem, there is a (random) proper subspace L ⊂ MD such that for X ∈ MD \ L we have 1 lim ln Φ (n) (X) = λ . n→∞ n Furthermore from 5.2.1 we see that l = limn n1 ln Φ (n) (X) for X ∈ SD and as SD spans MD it follows from Oseledet’s Theorem that l = λ. Now we proceed to prove all the necessary results in the previous discussion. We present the following lemmas: Lemma 5.4.2. Under assumption I the random sequence (Dn )n≥1 defined in 5.2.3 is eventually bounded with probability 1. That is almost surely Dn (ω) is eventually bounded. Proof. We restrict our attention to the probability 1 event where τ < ∞. For such ω we can find n ∈ N such that n > τ(ω). Then for any X, Y ∈ SD we have that Φ (n) (X) is positive definite. Therefore we have 0 < ⟨Y , Φ (n) (X)⟩ = ⟨φτ+1 ∗ ◦ . . . ◦ φn∗ (Y ), Φ (τ) (X)⟩. (5.4.4) ∗ Let W = φτ+1 ◦ . . . ◦ φn∗ (Y ). Then we have that; 0 < ⟨W , Φ (τ) (X)⟩ ≤ ∥W ∥∞ Φ (τ) (X) ≤ ∥W ∥ Φ (τ) . (5.4.5) The second inequality follows from the Hölder’s inequality for Schatten norms and the last inequality follows from the fact that Schatten p- norms are non-increasing in p. there- fore ln⟨Y , Φ (n) (X)⟩ ≤ ln ∥W ∥ + ln Φ (τ) . (5.4.6) Now since Φ (τ) is strictly positive, then we must have that for each nonzero rank one projection, P , the map P 7→ min σ (Φ (τ) (P )) is continuous and takes values on (0, ∞). Here σ denotes the spectrum. Now as the set of projections is a compact set we have that there exists a > 0 such that a = min{min(σ (Φ (τ) (P ))) : P is a rank 1 projection}. Now for X ∈ SD 94 we have that there is a rank one projection P such that, X ≥ ∥X∥∞ P . Therefore we have that a ⟨W , Φ (τ) (X)⟩ ≥ ⟨W , ∥X∥∞ Φ (τ) (P )⟩ ≥ a∥X∥∞ ⟨W , I⟩ = a∥X∥∞ ∥W ∥ ≥ ∥W ∥. (5.4.7) D The last equality uses that W is positive semi-definite and therefore its trace is equal to its trace norm. W is positive semi-definite, because φi , hence φi∗ are positive maps, for all i. This gives us that a ln⟨Y , Φ (n) (X)⟩ ≥ ln + ln ∥W ∥. (5.4.8) D As for n > τ we have ln⟨Y , Φ (n) (X)⟩ , −∞, combining 5.4.8 and 5.4.6 we see that; n a o ln⟨Y , Φ (n) (X)⟩ − ln φτ+1 ∗ . . . ◦ φn∗ (Y ) ≤ max ln , ln Φ (τ) . (5.4.9) D 1 Now as 5.4.9 holds for any X, Y ∈ SD , we can substitute X = DI in to 5.4.9. This gives us that; 1 n a o ln Φ (n)∗ (Y ) − ln φτ+1 ∗ . . . ◦ φn∗ (Y ) ≤ max ln , ln Φ (τ) . (5.4.10) D D Combining 5.4.9 and 5.4.10, we see that in the event that τ < ∞, Dn (ω) is eventually bounded. Since Dn is eventually bounded almost surely, we immediately have the following limiting properties of (Dn )n∈N . Dn D Corollary 5.4.3. For Dn defined above we have that lim and lim √ n , converge to 0 almost n→∞ n n→∞ n surely. Equipped with the previous lemma (and the corollary) we are ready to prove that lim n → ∞(1/n)En = 0 almost surely. Lemma 5.4.4. Under assumptions I for the random sequence (En )n≥1 defined above 5.2.4 we have that the sequence (En (ω)/n)n≥1 converges to 0 almost surely. 95 Proof. Let ω be such that τ(ω) < ∞ and choose n > τ(ω). Let X, Y ∈ SD . Consider (n)∗ Φω (Y ) . We have that; n−1 X (n)∗ ln Φ (Y ) = ln ϕk∗ ((ϕk+1 ∗ ◦ . . . ◦ ϕn∗ ) · Y ) + ln ∥ϕn∗ (Y )∥. (5.4.11) k=1 Now let α ∈ (0, 1) and let nα be the integer part of (1 − α)n. Then we have that X n (n)∗ 1 2 ln Φ (Y ) − ln ϕk∗ (Zk+1 ) ≤ Sn,ω (Y ) + Sn,ω (Y ), where (5.4.12) k=1 Xnα Sn1 (Y ) = ln ϕk∗ ((ϕk+1 ∗ ◦ . . . ◦ ϕn∗ ) · Y ) − ln ϕk∗ (Zk+1 ) , (5.4.13) k=1 and Xn−1 Sn2 (Y ) = ln ϕk∗ ((ϕk+1∗ ◦ . . . ◦ ϕn∗ ) · Y ) − ln ϕk∗ (Zk+1 ) k=nα +1 + ln ∥ϕn∗ (Y )∥ − ln ∥ϕn∗ (Zn+1 )∥ . (5.4.14) From lemma 2.7.4 (which can be applied because of assumption I) we have that nα nα X ϕk∗   X ϕk∗  ∗  Sn1 (Y ) ≤ 2 d (ϕ ∗ ◦ . . . ◦ ϕ ∗ n ) · Y , Z k+1 ≤ 2 c (ϕ ◦ . . . ◦ ϕ ∗ n ) v(ϕk∗ ) k+1 v(ϕk∗ ) k+1 k=1 k=1 nα X ϕk∗   ≤2 c (ϕn∗ α +1 ◦ . . . ◦ ϕn∗ ) . (5.4.15) v(ϕk∗ ) k=1 Second to last inequality uses lemma 4.2.1 and the last inequality uses lemma 2.8.1. Now since the right hand side of the above inequality is independent of the choice of Y we have nα X ϕk∗   Sn1 = sup S1 (Y ) ≤ 2 c (ϕn∗ α +1 ◦ . . . ◦ ϕn∗ ) . (5.4.16) Y∈ SD v(ϕk∗ ) k=1 We now proceed to prove that lim supn→∞ n1 ln Sn1 < 0. If that were the case we must ln κ have that limn→∞ Sn1 = 0. To this end let 0 < ϵ < − α1−α . (Such ϵ exists as, − ln κ ∈ (0, ∞) and   α ∈ (0, 1)). Now by assumption I we must have that E[ln ϕ0∗ /v(ϕ0∗ ) ] is finite. Therefore 96 we have that X     X     ∞> P ln ϕ0∗ /v(ϕ0∗ ) > kϵ = P ln ϕk∗ /v(ϕk∗ ) > kϵ (5.4.17) k≥1 n≥1 X  ϕ∗  k kϵ = P > e . (5.4.18) v(ϕk∗ ) k≥1 Thus by Borel-Cantelli lemma we have that  ϕ∗  k −kϵ P e > 1 for infinitely many k = 0. (5.4.19) v(ϕk∗ ) This allows us to define an almost sure finite function C as ϕk∗ (ω) C(ω) = sup e−kϵ . (5.4.20) k≥1 v(ϕk∗ (ω)) Now going back to S1 and taking further restricting ω to satisfy that C(ω) < ∞ we see that Xnα   0 ≤ S1 (ω) ≤ 2C(ω) c (ϕn∗ α +1 ◦ . . . ◦ ϕn∗ )(ω) k=1 eϵ   ≤ 2C(ω) eϵnα c (ϕn∗ α +1 ◦ . . . ◦ ϕn∗ ) . (5.4.21) eϵ − 1 Now taking logarithm and applying lemma 5.3.1 with proposition 2.8.1, gives us that 1 n 1   lim sup ln Sn1 (ω) ≤ lim ϵ α + lim ln c (ϕn∗ α +1 ◦ . . . ◦ ϕn∗ ) = ϵ(1−α)+α ln κ < 0. (5.4.22) n→∞ n n→∞ n n→∞ n Therefore we must have that limn→∞ Sn1 = 0 almost surely. Sn2 Now we proceed to prove that limn→∞ n = 0 almost surely. We first start by observing that by assumption I, ker(ϕ0∗ ) ∩ SD is a.s. empty. Therefore we have that a.s. for all n, ker(ϕn∗ ) ∩ SD is empty. Therefore a.s. for all n we must have that | ln ∥ϕn∗ (X)∥ | ≤ | ln v(ϕn∗ )| + | ln ∥ϕn∗ ∥ | for all x ∈ SD (5.4.23) X n Hence Sn1 (Y ) ≤ 2 | ln v(ϕn∗ )| + | ln ∥ϕn∗ ∥ | (5.4.24) k=nα +1 By assumption I and the Birkhoff ergodic theorem [6] we then get that for Sn1 = sup Sn1 (Y ) Y ∈SD 97 Sn1 lim supn→∞ n ≤ 2α[E[ ln ϕ0∗ ] + E[ ln v(ϕ0∗ ) ]]. Since α ∈ (0, 1) was chosen arbitrarily we Sn1 get the desired result. Now combining the convergences Sn2 → 0 almost surely and n →0 almost surely we have that n 1 1X lim sup ln Φ (n)∗ (Y ) − ln ϕk∗ (Zk+1 ) converge to 0 a.s. (5.4.25) n→∞ Y ∈S n n D k=1 Combining 5.4.25 with lemma 5.4.2 (corollary 5.4.3) we finish the proof of lemma 5.4.4. With the discussion in the beginning of this section we see that the lemmas together prove the Law of Large Numbers Theorem. 98 CHAPTER 6 CLT FOR ERGODIC QUANTUM PROCESSES 6.1 Filtration and Martingales In this section we propose a central limit type theorem for ergodic quantum processes in an invertible ergodic system as defined in 3.3.6. To obtain a central limit type theorem we require the following integrability assumption: Assumption I(p). For p ≥ 1, the random variables ln ∥φ0∗ ∥ and ln v(φ0∗ ) are in Lp (Ω) We state the proposed Central Limit Theorem (CLT) below. Remark 6.1.1. Since for a finite measure space (X, X , µ) we have that for 1 ≤ p < q ≤ ∞, Lq (X, X , µ) ⊂ Lp (X, X , µ) it follows that under assumption I(p) for any p ≥ 1 the random vari- ables ln ∥φ0∗ ∥ and ln v(φ0∗ ) are are in L1 . Central Limit Theorem 6.1.2. Let (φ(n) )n∈Z be a random ergodic sequence of positive maps satisfying assumptions B’1 and B’2. Let Φ (n) be defined as in 3.3.6. If the assumption I(p) holds for some p ≥ 2 then we have that: If X∞ ∥E[ξ0 |F n ]∥q < ∞ (6.1.1) n=1 with 1/p + 1/q = 1, then for any sequences (Xn )n≥1 and (Yn )n≥1 in Sn , the random sequence ! 1   √ ln⟨Yn , Φ (n) (Xn )⟩ − nl (6.1.2) n n≥1 converges in distribution to a centered normal random variable with variance  2 " X ! #  2 0 1 σ := E E[ξ−k |F ] − E[ξ−k |F ] ≥ 0. (6.1.3) k≥0 The center of the proof of the CLT is an application of the martingale approximation method, also know as Gordin’s method [30]. We shall also show that the summability in 6.1.1 is satisfied if the system if exponentially ρ-mixing (for the case p=2) or if the system is exponentially α-mixing (for the case p > 2) in the next section. To understand these we need the notion of a filtration and martingales. 99 Definition 6.1.3 (Forward and Reverse Filtrations). Let (Ω, F ) be a measurable space and let (An )n∈Z be a sequence of sub σ -algebra. If An ⊆ An+1 for all n ∈ Z the collection (An )n∈N is called a forward filtration. But if An+1 ⊆ An for all N ∈ Z the collection (An )n∈Z is called a backward (reverse) filtration. With the above definition we introduce the natural forward and backward filtrations generated by the sequence (φn )n∈Z Fn = σ {φk : k ≤ n}. (6.1.4) F n = σ {φk : k ≥ n}. (6.1.5) We recall from chapter 1 that σ (A) denotes the σ -algebra generated by set A (i.e. the inter- section of all σ -algebras containing A) and for a collection of measurable functions (Xt )t∈T taking values in some measurable space (S, S), we define σ ((Xt )t∈T ) := σ (∪t∈T {Xt−1 (S)}). We readily see that (Fn )n∈N is a forward filtration and (F n )n∈N is a reverse filtration. Definition 6.1.4 (Martingales and Martingale Increments). Let F := (Fn )n∈N be a (forward) filtration then a process X := (Xn )n∈N taking values in some measurable space (S, S) is called a F -martingale if 1. Xn is (S, Fn )-measurable and 2. For all m ≤ n we have E[Xn |Fm ] = Xm almost surely. Here E[ · | · ] denotes the conditional expectation defined in the theorem 2.1.6. If the process instead satisfies that 1. Xn is (S, Fn )-measurable and 2. For all m < n we have E[Xn |Fm ] = 0 almost surely, then X is called a F - martingale increment (or martingale difference). Similarly if F ′ := (F n )n∈N is a reverse filtration a process Y := (Yn )n∈N taking values in some measurable space (S, S) is called a F ′ - reverse martingale if 1. Yn is (S, F n )-measurable and 100 2. For all m ≤ n we have E[Xm |Fn ] = Xn almost surely. If the process Y satisfies the following conditions: 1. Yn is (S, F n )-measurable. 2. For all m < n we have E[Xm |Fn ] = 0 almost surely. Then Y is called a F ′ reverse martingale difference (reverse martingale increment). For martingale differences we have a central limit theorem [4, 10, 15] which was proved independently by Billingsly [5] and Ibragimov [37] for the ergodic case. Martingale Difference Central Limit Theorem 6.1.5. Let (Xn )n≥1 be a stationary ergodic direct or reversed martingale difference with respect to a filtration {An }n≥1 . If X1 ∈ L2 , then σ 2 = E(X12 ) and the Central Limit Theorem holds for the sequence (Xn )n≥1 . 6.2 Proof of the Central Limit Theorem The following proof is adapted from the proof of [34, Lemma 9.2] and is similar to the proof of [52, Theorem 1.1]. The key idea in the proof of the central limit theorem is to approximate the following quantity via reverse martingale differences ln (ϕk∗ (ω))(Zk+1 (ω)) − l then to prove that n 1 X  √ ln (ϕk∗ (ω))(Zk+1 (ω)) − l n k=1 converges in distribution to a centered normal law using the central limit theorem for martingale differences. Since for any sequences (Xn )n≥1 and (Yn )n≥1 in Sn we have that 1   √ 1τ E[| ln ϕ0∗ | + | ln v(ϕ0∗ )|]. (6.2.6) 102 Thus for all ϵ > 0, | ln ϕ0∗ | + | ln v(ϕ0∗ )| | ln ∥ϕn∗ ∥| + | ln v(ϕn∗ )| √ ! X  X √  ∞> P > n = P > n (6.2.7) ϵ ϵ n≥1 n≥1 Here the last equality comes from the fact that θ is measure preserving. Now by Borel- Cantelli lemma we have that | ln ∥ϕn∗ ∥| + | ln v(ϕn∗ )|   P lim sup √ >ϵ =0 (6.2.8) n→∞ n Therefore | ln ∥ϕn∗ ∥| + | ln v(ϕn∗ )| √ → 0 a.s. (6.2.9) n and thus √1 Sn0 → 0 almost surely as n → ∞. n Now we shall prove that Sn1 converges in distribution to some S ′ . We notice that this d d will yield the desired result as, if Sn1 → S ′ (here → denotes the convergence in distribut- ing) we must have that √1 Sn1 → 0 in distribution (see for an example Slutsky’s theorem) n and thus to 0 in probability. To this end let (Sn1 )′ := Sn1 ◦ θ −(n+1) . Then we have that n−1 ∗ X ϕ−l−1   (Sn1 )′ = ∗ c ϕ−l ∗ ∗ ◦ . . . ◦ ϕ−1 . (6.2.10) v(ϕ−l−1 ) l=1 We apply the root test for the infinite series given by X ϕ∗ X −l−1   ∗ ∗ ∗ c ϕ−l ◦ . . . ◦ ϕ−1 := al . (6.2.11) v(ϕ−l−1 ) l≥1 l≥1   ∗ ∗ Now as ln ϕ−l−1 /v(ϕ−l−1 ) are identically distributed in l, we have that (using Borel- Cantelli lemma, similar to the proof of 6.2.8) for each ϵ > 0  ∗  1  ϕ−l−1  ! P lim sup ln  ∗ ≥ϵ =0 (6.2.12) l→∞ l v(ϕ−l−1 ) Since the above holds for all ϵ > 0 we have that  ∗  1  ϕ−l−1  ! P lim sup ln  ∗ =0 =1 (6.2.13) l→∞ l v(ϕ−l−1 ) 103   Furthermore by corollary 3.3.9 we also have that liml→∞ ln 1l c ϕ−l ∗ ◦ . . . ϕ ∗ −1 = ln κ′ < 0. Therefore the series in 6.2.11 converges a.s. So there is some (alsmot sure finite) S ′ such that as n → ∞, a.s d d Sn1 d (Sn1 )′ → S ′ =⇒ (Sn1 )′ → S ′ =⇒ Sn1 → S ′ =⇒ √ → 0. (6.2.14) n Finally, as convergence to a constant in distribution implies the convergence in probabil- S1 ity we have that limn→∞ √nn → 0 in probability, finishing the proof. Lemma 6.2.2. Let p ≥ 2 and let q be such that 1/p + 1/q = 1. Then assume that the random variables defined by w 7→ ln ϕ0∗ (ω) and ω 7→ ln v(ϕ0∗ (ω0 )) are in Lp . Further assume that X∞ ∥E[ξ0 |F n ]∥q < ∞. (6.2.15) n=1 Then we have that the sequence n ! 1 X √ ξk (6.2.16) n k=1 n≥1 converge in distribution to a centered normal law with variance σ 2 . Here we allow σ = 0 possibility and σ = 0 if and only if there exists stationary sequence (gn )n≥1 such that gn ∈ Lq (F n ) and ξn = gn+1 − gn . (6.2.17) |E[ξ−k |F 0 ]| . Because of 6.2.15 we must have that P Proof. First consider the series k≥1 this series converges in L1 and hence absolutely on an event of probability one. This can be established by the following inequalities; X X X 0 0 E[ξ−k |F ] 1 ≤ E[ξ−k |F ] q = E[ξ0 |F k ] q < ∞. (6.2.18) k≥1 k≥1 k≥1 Because we also have that w 7→ ln ϕ0∗ (ω) in Lp and as p ≥ 2 we also have that ξn ∈ L2 for all n ≥ 0. This means that for X g0 = E[ξ−k |F k ] k≥1 we have that X ∥g0 ∥q ≤ E[ξ0 |F k ] q = M < ∞. k≥1 104 Now in the event where g0 = P 0] converges absolutely, define k≥0 E[ξ−k |F X  X ζ0 = E[ξ−k |F 0 ] − E[ξ−k |F 1 ] and g0 = E[ξ−k |F 0 ]. (6.2.19) k≥0 k≥1 And let ζn = ζ0 ◦ θ n and gn = g0 ◦ θ n for all n ∈ Z. Since g0 ∈ Lq (F 0 ), we have that gn ∈ Lq (F n ). We also have that ζn , gn ∈ F n and that; ξn = ζn + gn+1 − gn . (6.2.20) Now taking conditional expectation with respect to F n+1 in 6.2.19, we see that; E[ζn |F n+1 ] = 0. (6.2.21) This means that (ζn )n≥1 is a reverse martingale difference. Now 6.2.20 shows us that n n 1 X 1 X 1 √ ξk = √ ζn + √ (gn+1 − g1 ). (6.2.22) n k=1 n k=1 n Now as limn→∞ √1n (gn+1 − g1 ) converges to 0 in probability, we have the required con- vergence in distribution if we establish the central limit theorem for the martingale dif- ference (ζn )n≥1 . As a courtesy of the CLT for martingale differences we only need to prove that ζn is square integrable for all n ≥ 1. Since ξn are square integrable, we have that ζn is square integrable if and only of gn+1 − gn is square integrable. As gn is stationary, we only need to prove that g1 − g0 is square integrable. Let λ ∈ (0, 1) and in the event where k≥0 E[ξ−k |F 0 ] converges absolutely, define P X g0λ = λk−1 E[ξ−k |F 0 ] and gnλ = g0λ ◦ θ n for n ∈ Z. (6.2.23) k≥1 2 Now we proceed to find an upper bound for g1λ − λg0λ 2 . 2 g1λ − λg0λ 2 = E[(g1λ )2 − 2λg1λ g0λ + λ2 (g0λ )2 ] (6.2.24) 2 2 = 2 g1λ 2 − 2λE[g0λ g1λ ] − (1 − λ2 ) g0λ 2 (6.2.25) h 2 i ≤ 2 g1λ 2 − λE[g0λ g1λ ] (6.2.26) h 2 i = 2 g1λ 2 − λE[E[g0λ g1λ |F 1 ]] (6.2.27) h 2 i = 2 g1λ 2 − λg1λ E[g0λ |F 1 ] . (6.2.28) 105 The second equality above uses that as (gnλ )n∈N are stationary we have that g1λ 2 = g0λ 2 . 2 The inequality above uses that (1 − λ2 ) g0λ 2 ≤ 0. Last equality uses that g1λ is F 1 measur- able. Also note that X X g1λ − λE[g0λ |F 1 ] = λk−1 E[ξ−k+1 |F 1 ] − λ λk−1 E[ξ−k |F 1 ] = E[ξ0 |F 1 ] (6.2.29) k≥1 k≥1 Therefore we have that; Z λ 2 λ g1 − λg0 2 ≤ 2 E[ξ0 |F 1 ]g1λ dP ≤ 2 E[ξ0 |F 1 ] p g1λ q ≤ 2∥ξ0 ∥p M. (6.2.30) Ω The second to last inequality is an application of Hölder’s inequality and the last inequal- ity uses that λ < 1 and therefore goλ q ≤ M. Now as right hand-side of above inequalities are independent of λ we have that; 2 sup g1λ − λg0λ 2 < ∞. (6.2.31) λ∈(0,1) This gives us that h i 2 E[(g1 − g0 )2 ] = E lim inf− (g λ 1 − λg λ 2 0 ) ≤ lim inf − E[(g1λ − λg0λ )2 ] < sup g1λ − λg0λ 2 < ∞, λ→1 λ→1 λ∈(0,1) (6.2.32) by Fatou’s lemma. Hence (g1 − g0 ) is square integrable which in return gives us that ζn is square integrable for each n and thus CLT for martingale differences applies. CLT for martingale differences also gives us that the variance σ 2 = E[ζ02 ]. If σ = 0 then we have that ζ0 and hence ζn for each n ∈ Z is 0 a.s. (as ζ0 is centered). Therefore we must have that ξn = gn+1 − gn for the stationary processes (gn )n∈Z defined above. This concludes the proof of lemma 6.2.2. 6.3 Mixing Conditions In this section we present properties that are related to mixing coefficient (defined be- low) to guarantee the convergence of 6.1.1 that was required for the central limit theorem 6.1.2. The arguments in this section are based on similar results in [20], [14] and [34]. We introduce the following mixing coefficients for two sub σ -algebras on a probability space (Ω, F , P). 106 Definition 6.3.1. Let (Ω, F , P) be a probability space and let A, B be two sub σ -algebras. Then we define the following two measures of the dependence; α(A, B) = sup{|P(A ∩ B) − P(A) − P(B)| : A ∈ A, B ∈ B}. ( ) E[(Y − E[Y ])(X − E[X])] 2 2 ρ(A, B) = sup : Y ∈ L (A), X ∈ L (B), X, Y , 0 . σ (Y )σ (X) We note that the two σ -algebras are independent if and only if α(A, B) = 0 if and only if ρ(A, B) = 0. ρ(A, B) is called the “maximal correlation” between the σ -fields A and B. With the definition of previous two measures of dependence we define the α and ρ mixing coefficients of a sequence of random variables as follows. We also refer the reader to [8] for a survey on mixing coefficients or to [9] for a detailed treatment of theses mixing coefficients. Definition 6.3.2. Let (Xn ) be a sequence of random variables and let Fk = σ (Xn : n ≤ k) and F k = σ (Xn : n ≥ k). Then we define αn := sup α(Fk , F n+k ) (6.3.1) k∈N ρn := sup ρ(Fk , F n+k ) (6.3.2) k∈N Remark 6.3.3. Historically if α(n) → 0 as n → ∞ the random sequence (fn )n∈Z is called “strongly mixing” and if ρ(n) → 0 as n → ∞ the random sequence (fn )n∈Z is called “ρ-mixing”. The strong mixing condition α(n) → 0 was introduced in [62]. The mixing condition ρ(n) → 0 was introduced in [47]. As we shall see later in theorem 6.3.8 to guarantee the convergence in 6.1.1 we require either exponentially strongly mixing or exponentially ρ-mixing (depending on the value of p in assumption I(p)) as described in 6.3.8. We have the following properties for αn and ρn . Lemma 6.3.4. [18, §1.2 Theorem 3] Let n, k ∈ N then for p, q, r ∈ [1, ∞] such that 1/p + 1/q + 1/r = 1, we have that for each X ∈ Lp (Fk ) and Y ∈ Lq (F k+n ) we have E[XY ] − E[X]E[Y ] ≤ 8(α(Fk , F n+k ))1/r ∥X∥p ∥Y ∥q (6.3.3) 107 And for X ∈ L2 (Fk ) and Y ∈ L2 (F n+k ), we have E[XY ] − E[X]E[Y ] ≤ ρn (Fk , F n+k ) ∥X∥2 ∥Y ∥2 . (6.3.4) We also have the following results, proofs of which can be found on [34, lemma 6.2, lemma 6.3]. Lemma 6.3.5. Let (Mn )n≥1 be a [0, 1]-valued sequence of random variables on (Ω, F , P, θ) where θ is an ergodic measure preserving transformation. Suppose that (Mn )n≥1 has the fol- lowing sub-multiplicative property Mm+n ≤ Mm Mn ◦ θ n . (6.3.5) If for 0 ≤ m < n we have that Mn−m ◦ θ m is both Fn and F m measurable then we have: 1. Assume that for some λ > 0 and for some c we have that αn ≤ c/8nλ . Then for any sequence (an )n≥1 of real numbers such that ln n a lim = lim n = 0 (6.3.6) n→∞ an n→∞ n there exists c′ such that for all n ∈ N, !λ a E[Mn ] ≤ c n ′ . (6.3.7) n 2. Assume that limn→∞ ρn = 0. Then for each k ∈ N, there exists c′′ ∈ R such that for each n ≥ 1, c′′ E[Mn ] ≤ . (6.3.8) nk Corollary 6.3.6. Suppose that assumption B’1 and B’2 hold then for r ∈ (0, 1) define;     τr = inf{n ≥ 1 : c Ψ (n) , c Φ (n) ≤ r}. (6.3.9) Where Ψ (n) = φ−n ∗ ◦ . . . ◦ φ∗ Then we have that τ < ∞ a.s. Moreover −1 r 1. If for some λ > 0, we have k≥1 αk1/λ < ∞, then there exists a constant K such that for all P n≥1 2 !2   ln n max{P[τr > n], E[c Φ (n) ]} ≤ K . (6.3.10) n 108 2. If limn→∞ ρn = 0, then there exists a constant K ′ such that for all n ≥ 1   1 max{P[τr > n], E[c Φ (n) ]} ≤ K ′ 8 . (6.3.11) n Proof. From lemma 3.3.7 we directly get the result that P[τr < ∞] = 1. We also have by assumption B’1 that a.s. for all n ≥ 1, SD ∩ ker(φn ) = ∅. Therefore a.s. we have that for all n ∈ Z, c (Ψ τr +n ) , c (Φ τr +n ) < r by lemma 2.8.1. Case 1: For the first part, we start with the observation that αn is decreasing in n, this can be seen by the definition of αn in 6.3.1 using the fact that (F n )n≥1 is decreasing in n. Therefore (αnλ )n∈N is a decreasing sequence of non negative real numbers such that n∈N αnλ converges. Thus we have that limn→∞ nαnλ = 0. Thus the sequence nαnλ P is bounded and therefore we can find a constant c such that for all n ≥ 1, αn ≤ cn−λ . Now notice that (c(Φ (n) ))n∈N satisfies the sub-multiplicative condition in lemma 6.3.5, with Mn = c(Φ (n) ). We also see that for the sequence whose general term is defined by an = ln2 n. we must have that limn→∞ an /n = limn→∞ ln n/an = 0. Therefore the hypothesis in part one of lemma 6.3.5 applies and we get that !λ   ln2 n E[c Φ (n) ] ≤ c1 . (6.3.12) n Preceding analysis can also be applied to (c (Ψ n ))n∈N . Therefore we can find a constant c2 such that, !λ   ln2 n E[c Ψ (n) ] ≤ c2 . (6.3.13) n We also have that 1 1 P[τr > n] ≤ P[c(Ψ (n) ) > r] + P[c(Φ (n) ) > r] ≤ E[c(Φ (n) )] + E[c(Ψ (n) )]. (6.3.14) r r Therefore we can find a constant K such that 6.3.10 holds. Case 2: Now when limn→∞ ρn = 0, the second part of lemma 6.3.5 applies. We still have that 6.3.14 holds. These two combined gives us 6.3.11. 109 Lemma 6.3.7. Suppose that the assumptions B’1, B’2 and I(p) hold. Let r ∈ (0, 1) and let τr be defined as in 6.3.6. Let nα denote the integer part of (1 − α)n, for α ∈ (0, 1). 1. If for some δ > 0 then m2+δ < ∞ we have that δ/(2+δ)   ∥E[ξ0 |F n ]∥(2+δ)/(1+δ) ≤ 8αn−nα (4m2+δ + 2r ′ ) + 4r ′ E[c Φ (nα )∗ ] + 4m2+δ (P[τr > nα ])(1+δ)/(2+δ) . (6.3.15) 2. If m2 ≤ ∞ then we have that   ∥E[ξ0 |F n ]∥2 ≤ ρn−nα (4m2 + 2r ′ ) + 4r ′ E[c Φ (nα )∗ ] + 4mp (P(τr > nα ))(1/2) . (6.3.16) Here r ′ is the (deterministic) real-number r ′ = 1r ln 1−r 1 . Proof. Observe that by the definition of τr we must have that τr ≥ τ. By lemma 4.2.1 we can obtain that Φ (nα ) · Znα +1 = Z1 . Therefore we have that ξ0 = An + Bn − E[An ]. (6.3.17) Where An = ln φ0∗ (Φ (nα ) · Znα +1 ) − ln φ0∗ (Φ (nα ) · I/D) (6.3.18) and Bn = ln φ0∗ (Φ (nα ) · I/D) − E[ln φ0∗ (Φ (nα ) · I/D) ]. (6.3.19)   Now consider the event {ω : τr ≤ nα }. Then we have that on this event, c Φ (nα )∗ ≤ r almost surely. This is due to assumption B’1 implying the property that ker(φ0∗ ) ∩ SD = ∅,     and therefore by lemma 2.8.1, c Φ (n) = c Φ (n)∗ a.s. Now from lemma 2.8.3 and by the assumption I(p) we have that  2 1   |An | ≤ A′n = c Φ (nα )∗ ln + 2 | ln φ0∗ | + | ln v(φ0∗ )| 1τr >nα a.s. (6.3.20) r 1−r 110 This gives us that   2 1 E|An | ≤ E|A′n | ≤ E[c Φ (nα )∗ ] ln + 2mp (P(τr > nα ))(p−1)/p . (6.3.21) r 1−r Furthermore we also have that 2 1 A′n p ≤ ln + 2mp and ∥B∥p ≤ 2mp . (6.3.22) r 1−r Now, for 1/p + 1/q = 1 we have that Z n ∥E[ξ0 |F ]∥q = sup f E[ξ0 |F n ] dP (6.3.23) {f ∈Lp (F n ):∥f ∥ p =1} Z = sup E[f ξ0 |F n ] dP (6.3.24) {f ∈Lp (F n ):∥f ∥p =1} = sup E[f ξ0 ] . (6.3.25) {f ∈Lp (F n ):∥f ∥p =1} Hence to prove that ∥E[ξ0 |F n ]∥q < ∞ we find a uniform upper bound for {|E[ξf ]| : f ∈ Lp (F n ) and ||f ||p = 1}. Now as ξo = An + Bn − E[An ] we have that, |E[ξ0 f ]| ≤ |E[An f ]| + |E[Bn f ]| + |E[An ]E[f ]| (6.3.26) ≤ E[A′n |f |] + |E[Bn f ]| + E[A′n ]E[|f |] (∵ A′n ≥ 0) (6.3.27) ≤ E[A′n |f |] − E[A′n ]E[|f |] + 2E[A′n ]E[|f |] + |E[Bn f ]| (6.3.28) = E[A′n |f |] − E[A′n ]E[|f |] + 2E[A′n ]E[|f |] + |E[Bn f ] − E[Bn ]E[f ]||. (6.3.29) Last equality uses that E[Bn ] = 0. Now we apply lemma 6.3.4 for the cases p = 2 + δ and p = 2 separately. Case 1: For p = 2 + δ, let q = 2 + δ and r = (2 + δ)/δ. Notice that A′n , Bn ∈ L2+δ (Fnα ), and f ∈ L2+δ (F nα +(n−nα ) ). Therefore whenever ∥f ∥2+δ = 1 we get, δ/(2+δ) |E[ξ0 f ]| ≤ 8αn−nα ( A′n 2+δ + ∥Bn ∥2+δ )∥f ∥2+δ + 2E[A′n ]∥F∥2+δ (6.3.30) ! δ/(2+δ) 2 1 ≤ 8αn−nα 4m2+δ + ln + 4m2+δ (P[τr > nα ])(1+δ)/(2+δ) r 1−r 4 1   + ln E[c Φ (nα )∗ ]. (6.3.31) r 1−r 111 Case 2: For p = 2, let q = 2. We have that A′n , Bn ∈ L2 (Fnα ), and f ∈ L2 (F nα +(n−nα ) ) Then by the second part of lemma 6.3.4 we have that for ∥f ∥2 = 1;   |E[ξ0 f ]| ≤ ρn−nα A′n 2 + ∥B n ∥ + 2E[A′n ]∥F∥2+δ (6.3.32) 2 1   4 1 ≤ ρn−nα (4m2 + ln ) + E[c Φ (nα )∗ ] ln r 1−r r 1−r + 4mp (P(τr > nα ))(1/2) . (6.3.33) Therefore by the discussion above, we have the desired results. Now we give properties of mixing coefficients to guarantee the convergence of the series in 6.1.1 which in return gives us the CLT. P δ/(2+δ) Theorem 6.3.8. If there is δ > 0 such that m2+δ < ∞ and n≥1 αn < ∞ then X∞ ∥E[ξ0 |F n ]∥(2+δ)/(1+δ) < ∞. n=1 P If m2 < ∞ and n≥1 ρn < ∞ then we have that X ∞ ∥E[ξ0 |F n ]∥2 < ∞. n=1 Proof. Follows directly from corollary 6.3.6 and lemma 6.3.7. Remark 6.3.9. We note that if the sequence (ϕn )n∈Z is IID then the central limit theorem holds as αn = ρn = 0 trivially. Furthermore if (ϕn )n∈Z is a Markov chain such that there is some n P with ρn < 1 then we also have that n≥1 ρn < ∞ and thus CLT holds for p = 2 case. We refer P the reader to [9, 7.5 Theorem] for the summability of n≥ ρn . 112 BIBLIOGRAPHY [1] P. Arrighi and C. Patricot. “Conal representation of quantum states and non-trace- preserving quantum operations”. In: Physical Review A 68.4 (2003), p. 042310. [2] H. Bauer. Probability theory. Vol. 23. Walter de Gruyter, 2011. [3] R. Bhatia. Matrix analysis. Vol. 169. Springer Science & Business Media, 2013. [4] P. Billingsley. Probability and measure. John Wiley & Sons, 2008. [5] P. Billingsley. “The Lindeberg-Levy theorem for martingales”. In: Proceedings of the American Mathematical Society 12.5 (1961), pp. 788–792. [6] G. D. Birkhoff. “Proof of the ergodic theorem”. In: Proceedings of the National Academy of Sciences 17.12 (1931), pp. 656–660. [7] I. Bongioanni, L. Sansoni, F. Sciarrino, G. Vallone, and P. Mataloni. “Experimental quantum process tomography of non-trace-preserving maps”. In: Physical Review A 82.4 (2010), p. 042307. [8] R. C. Bradley. “Basic properties of strong mixing conditions. A survey and some open questions”. In: Probability Surveys (2005), pp. 107–144. [9] R. C. Bradley. “Introduction to strong mixing conditions”. In: (2007). [10] B. M. Brown. “Martingale central limit theorems”. In: The Annals of Mathematical Statistics (1971), pp. 59–66. [11] L. Bruneau, A. Joye, and M. Merkli. “Random repeated interaction quantum sys- tems”. In: Communications in mathematical physics 284.2 (2008), pp. 553–581. [12] M.-D. Choi. “Completely positive linear maps on complex matrices”. In: Linear algebra and its applications 10.3 (1975), pp. 285–290. [13] D. L. Cohn. Measure theory. Vol. 1. Springer, 2013. [14] H. Cohn, O. Nerman, and M. Peligrad. “Weak ergodicity and products of random matrices”. In: Journal of Theoretical Probability 6.2 (1993), pp. 389–405. [15] J.-P. Conze and A. Raugi. “Limit theorems for sequential expanding dynamical systems”. In: Ergodic Theory and Related Fields: 2004-2006 Chapel Hill Workshops on Probability and Ergodic Theory, University of North Carolina Chapel Hill, North Carolina. Vol. 430. American Mathematical Soc. 2007, p. 89. 113 [16] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai. Ergodic theory. Vol. 245. Springer Science & Business Media, 2012. [17] J. L. Doob. Stochastic processes. John Wiley & Sons, 1953. [18] P. Doukhan. Mixing: properties and examples. Vol. 85. Springer Science & Business Media, 2012. [19] R. M. Dudley. Real analysis and probability. CRC Press, 2018. [20] D. Dürr and S. Goldstein. “Remarks on the central limit theorem for weakly depen- dent random variables”. In: Stochastic processes—Mathematics and physics. Springer, 1986, pp. 104–118. [21] R. Durrett. Probability: theory and examples. Vol. 49. Cambridge university press, 2019. [22] R. Engelking. “General topology”. In: Sigma series in pure mathematics 6 (1989). [23] D. E. Evans and R. Høegh-Krohn. “Spectral Properties of Positive Maps on C*- Algebras”. In: Journal of the London Mathematical Society 2.2 (1978), pp. 345–355. [24] S. N. Filippov. “Capacity of trace decreasing quantum operations and superaddi- tivity of coherent information for a generalized erasure channel”. In: Journal of Physics A: Mathematical and Theoretical 54.25 (2021), p. 255301. [25] S. N. Filippov. “Trace decreasing quantum dynamical maps: Divisibility and en- tanglement dynamics”. In: Infinite Dimensional Analysis, Quantum Probability and Applications: QP41 Conference, Al Ain, UAE, March 28–April 1, 2021. Springer. 2022, pp. 121–133. [26] D. H. Fremlin. Measure theory. Vol. 4. Torres Fremlin, 2000. [27] G. Frobenius, F. G. Frobenius, F. G. Frobenius, F. G. Frobenius, and G. Mathemati- cian. “Über Matrizen aus nicht negativen Elementen”. In: (1912). [28] H. Furstenberg and H. Kesten. “Products of random matrices”. In: The Annals of Mathematical Statistics 31.2 (1960), pp. 457–469. [29] I. Y. Gol’dsheid and G. A. Margulis. “Lyapunov indices of a product of random matrices”. In: Russian mathematical surveys 44.5 (1989), p. 11. [30] M. I. Gordin. “The central limit theorem for stationary processes”. In: Doklady Akademii Nauk. Vol. 188. 4. Russian Academy of Sciences. 1969, pp. 739–741. 114 [31] P. R. Halmos. Finite-dimensional vector spaces. Courier Dover Publications, 2017. [32] P. R. Halmos. Measure theory. Vol. 18. Springer, 2013. [33] P. R. Halmos and J. von Neumann. “Operator methods in classical mechanics, II”. in: Annals of Mathematics (1942), pp. 332–350. [34] H. Hennion. “Limit theorems for products of positive random matrices”. In: The Annals of Probability (1997), pp. 1545–1587. [35] E. Hille and R. S. Phillips. Functional analysis and semi-groups. Vol. 31. American Mathematical Soc., 1996. [36] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge university press, 2012. [37] I. Ibragimov. “A central limit theorem for a class of dependent random variables”. In: Theory of Probability & Its Applications 8.1 (1963), pp. 83–89. [38] C. T. Ionescu Tulcea. “Mesures dans les espaces produits”. In: Atti Accad. Naz. Lincei Rend 7 (1949), pp. 208–211. [39] A. Jamiołkowski. “Linear transformations which preserve trace and positive semidef- initeness of operators”. In: Reports on Mathematical Physics 3.4 (1972), pp. 275–278. [40] S. Kakutani. “32. Notes on Infinite Product Measure Spaces, I”. in: Proceedings of the Imperial Academy 19.3 (1943), pp. 148–151. [41] O. Kallenberg and O. Kallenberg. Foundations of modern probability. Vol. 2. Springer, 1997. [42] A. Kechris. Classical descriptive set theory. Vol. 156. Springer Science & Business Media, 2012. [43] J. L. Kelley. General topology. Courier Dover Publications, 2017. [44] J. F. Kingman. “Subadditive processes”. In: Ecole d’Eté de Probabilités de Saint-Flour V-1975. Springer, 1976, pp. 167–223. [45] J. F. C. Kingman et al. “Subadditive ergodic theory”. In: Annals of Probability 1.6 (1973), pp. 883–899. [46] A. Kolmogoroff. “Grundbegriffe der wahrscheinlichkeitsrechnung”. In: (1933). [47] A. N. Kolmogorov and Y. A. Rozanov. “On strong mixing conditions for stationary Gaussian processes”. In: Theory of Probability & Its Applications 5.2 (1960), pp. 204– 115 208. [48] K. Kraus. “General state changes in quantum theory”. In: Annals of Physics 64.2 (1971), pp. 311–335. [49] K. Kuratowski. Topology: Volume I. vol. 1. Elsevier, 2014. [50] C. W. Lamb. “A comparison of methods for constructing probability measures on infinite product spaces”. In: Canadian mathematical bulletin 30.3 (1987), pp. 282– 285. [51] T. M. Liggett. “An improved subadditive ergodic theorem”. In: The Annals of Prob- ability 13.4 (1985), pp. 1279–1285. [52] C. Liverani. “Central limit theorem for deterministic systems”. In: Pitman Research Notes in Mathematics Series (1996), pp. 56–75. [53] G. W. Mackey. “Borel structure in groups and their duals”. In: Transactions of the American Mathematical Society 85.1 (1957), pp. 134–165. [54] G. W. Mackey. “Point realizations of transformation groups”. In: Illinois Journal of Mathematics 6.2 (1962), pp. 327–335. [55] R. Movassagh and J. Schenker. “An ergodic theorem for quantum processes with applications to matrix product states”. In: Communications in Mathematical Physics 395.3 (2022), pp. 1175–1196. [56] R. Movassagh and J. Schenker. “Theory of ergodic quantum processes”. In: Physical Review X 11.4 (2021), p. 041001. [57] I. Nechita and C. Pellegrini. “Random repeated quantum interactions and random invariant states”. In: Probability Theory and Related Fields 152.1-2 (2012), pp. 299– 320. [58] J. Neveu. Mathematical foundations of the calculus of probability. Holden-day, 1965. [59] M. A. Nielsen and I. Chuang. Quantum computation and quantum information. 2002. [60] V. I. Oseledets. “A multiplicative ergodic theorem. Characteristic Ljapunov, expo- nents of dynamical systems”. In: Trudy Moskovskogo Matematicheskogo Obshchestva 19 (1968), pp. 179–210. [61] O. Perron. “Zur theorie der matrices”. In: Mathematische Annalen 64.2 (1907), pp. 248–263. 116 [62] M. Rosenblatt. “A central limit theorem and a strong mixing condition”. In: Pro- ceedings of the national Academy of Sciences 42.1 (1956), pp. 43–47. [63] O. Sarig. “Lecture notes on ergodic theory”. In: Lecture Notes, Penn. State University (2009). [64] S. M. Srivastava. A course on Borel sets. Vol. 180. Springer Science & Business Media, 2008. [65] Q. P. Stefano, I. Perito, and L. Rebón. “Selective and Efficient Quantum Process Tomography for Non-Trace-Preserving Maps: Implementation with a Supercon- ducting Quantum Processor”. In: Physical Review Applied 19.4 (2023), p. 044065. [66] W. F. Stinespring. “Positive functions on C*-algebras”. In: Proceedings of the Ameri- can Mathematical Society 6.2 (1955), pp. 211–216. [67] T. Tao. An introduction to measure theory. Vol. 126. American Mathematical Soc., 2011. [68] J. Watrous. The theory of quantum information. Cambridge university press, 2018. [69] M. M. Wolf. “Quantum channels and operations-guided tour”. In: (2012). [70] K. Yamagata. “Quantum monotone metrics induced from trace non-increasing maps and additive noise”. In: Journal of Mathematical Physics 61.5 (2020), p. 052202. 117