PHD THESIS QUANTUM ENTROPIES, RELATIVE ENTROPIES, AND RELATED PRESERVER PROBLEMS DÁNIEL VIROSZTEK DEPARTMENT OF ANALYSIS BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS HUNGARY SUPERVISOR: PROF. DÉNES PETZ 2016 Contents Index of Notation 2 Chapter 1. Introduction 3 1. C ¤-algebras, von Neumann algebras 4 2. Tensor product 9 3. Continuous functional calculus 11 4. Commutative von Neumann algebras 13 Chapter 2. Quantum variances, generalized entropies and relative entopies 15 1. Decomposable quantum variances 16 2. Generalizations of the strong subadditivity inequality 21 3. Bregman divergences and their use 34 Chapter 3. Preserver problems 51 1. Maps preserving Bregman and Jensen divergences 53 2. Jordan triple endomorphisms of the positive cone 66 3. An application: the endomorphisms of the Einstein gyrogroup 78 Chapter 4. Summary 83 Ackowledgement 84 Bibliography 85 1 2 CONTENTS Index of Notation N the set of natural numbers Z the set of integers Q the set of rational numbers R the set of real numbers C the set of complex numbers x X x is an element of the set X 2 A B the Cartesian product of the sets A and B £ An the nth Cartesian power of the set A H a complex Hilbert space ­x, y® the inner product of the elements x and y of the Hilbert space H — we use the convention that the inner prod- uct is linear in the second variable and conjugate-linear in the first variable. B(H ) the set of bounded linear operators on H Bsa(H ) the set of self-adjoint linear operators on H BÅ(H ) the set of positive semidefinite linear operators on H BÅÅ(H ) the set of positive definite linear operators on H ran(.) the range of a linear operator ker(.) the kernel of a linear operator M the set of n n complex matrices n £ Msa the set of n n self-adjoint complex matrices n £ MÅ the set of n n positive semidefinite complex matrices n £ MÅÅ the set of n n positive definite complex matrices n £ [A]i,j the entry in the i-th row and j-th column of a matrix A Atr the transpose of the matrix A IA the identity element of the unital algebra A CHAPTER 1 Introduction The classical work [44] of Andrey Nikolaevich Kolmogorov laid the foundations of probability theory in 1933. In Kolmogorov’s approach, the basic concept of probability theory is the probability space. A probability space is a triplet (X ,A ,P), where X is an arbitrary set, A P(X ) is a σ- ⊆ algebra — P(X ) denotes the power set of X — and P is a finite measure on A which is normalized, that is, P(X ) 1. This means that a probabil- Æ ity space is nothing else but a measure space with total measure one, so one may consider probability theory as a branch of measure theory. On the other hand, probability theory is a richer structure than measure the- ory in the sense that several measure theoretical notions gain intuitive meanings from the viewpoint of a probability theorist. Without the re- quirement of generality, let us mention some of the intuitions which are associated with the notions of measure theory. The most basic concept is that the measurable sets — that is, the elements of the σ-algebra A — are considered to be events. A measurable function f :(X ,A ) (K,B) ! is called a real/complex random variable if K R or K C, respectively. R Æ Æ Therefore, the Lebesgue integral X f dP of the measurable function f is called the expected value — if it exists. As P is a finite measure, it is quite easy to guarantee the existence of the integral of a measurable function. ¡ ¯ ¯ ¢ If f is essentially bounded, that is, P {x X : ¯f (x)¯ K } 0 for some 2 È Æ K 0, then f is integrable, moreover, any power of f is integrable. This È R k latter fact is remarkable as the integral X f dP is called the kth moment of the random variable f and plays an important role in probability the- ory. Let us denote by L1 (X ,A ,P) the set of essentially bounded measur- able complex valued functions on the probability space (X ,A ,P). Let us introduce the notation ½ ¯ Z ¾ ¯ ¯ ¯2 L2 (X ,A ,P) f : X C¯ f is measurable and ¯f ¯ dP , Æ ! ¯ X Ç 1 as well. Clearly, L2 (X ,A ,P) is a Hilbert space with the inner product ­f ,g® R f gdP. Every bounded measurable function f : X C deter- Æ X 7! mines a bounded linear operator on the Hilbert space L2 (X ,A ,P) in the 3 4 1. INTRODUCTION following way. Set f L1 (X ,A ,P). Let us define the multiplication op- 2 erator M f by M : L2 (X ,A ,P) L2 (X ,A ,P), g M (g): f g. f ! 7! f Æ Straightforward computations show that M f is linear, and the proof of ¡ ¢ the boundedness of M is quite easy, as well. So, M B L2 (X ,A ,P) f f 2 for any f L1 (X ,A ,P). Moreover, the operator norm of M f coincides 2 ° ° ° ° with the supremum norm of f , that is, °M ° °f ° . This latter fact is f Æ also rather easy to prove. The map 1 ¡ 2 ¢ (1) M : L1 (X ,A ,P) B L (X ,A ,P) , f M ! 7! f is a canonical isometric embedding of the commutative normed algebra ¡ 2 ¢ L1 (X ,A ,P) into the normed algebra B L (X ,A ,P) , which is far from being commutative in general. This embedding is the starting point of the noncommutative generalization of probability theory. In the follow- ing section we give a brief introduction to the theory of von Neumann algebras, which are the appropriate mathematical objects to formalize the concepts of noncommutative probability theory. It is fair to remark that all the results of this thesis concern finite dimensional von Neumann algebras. 1. C ¤-algebras, von Neumann algebras DEFINITION 1 (Normed algebra). A unital complex algebra A en- dowed with the norm . is said to be a normed algebra, if the norm is k k submultiplicative, i. e., ab a b for any a,b A and the identity k k · k kk k 2 element is of norm one, that is, 1 1. k A k Æ The reader who claims that not only unital algebras are said to be normed algebras is right. However, any algebra which appears in this work is unital, so for the sake of convenience, we incorporated the re- quirement of unitality in the definition. DEFINITION 2 (Banach algebra). A normed algebra which is a Banach space — that is, a complete normed space — is called a Banach algebra. DEFINITION 3 (Involution). Let A be a complex algebra. A map : ¤ A A , a a¤ is called an involution if it satisfies the following proper- ! 7! ties. is antilinear: (¸a b)¤ ¸a¤ b¤ for any a,b A and ¸ C. ² ¤2 Å Æ Å 2 2 id, that is, (a¤)¤ a for any a A . ² ¤ Æ Æ 2 is an antihomomorphism with recpect to the product: (ab)¤ ² ¤ Æ b¤a¤ for any a,b A . 2 1. C ¤-ALGEBRAS, VON NEUMANN ALGEBRAS 5 DEFINITION 4 (C ¤-algebra). A Banach algebra endowed with an invo- 2 lution : A A which satisfies a¤a a for any a A is called a ¤ ! k k Æ k k 2 C ¤-algebra. The above definition of C ¤-algebras is rather abstract. However, we do not loose any generality if we consider the elements of a C ¤-algebra as bounded operators on an appropriate Hilbert space. Indeed, any C ¤- algebra is isomorphic to a closed (in the operator norm topology) unital *-subalgebra (that is, it is closed under the involution) of the operator al- gebra B(H ) for a suitable Hilbert space H . Furthermore, any commuta- tive C ¤-algebra is isomorphic to C(X ) for some compact Hausdorff space X . (The symbol C(X ) denotes the algebra of all continuous complex- valued functions defined on X endowed with the supremum norm.) Despite the above remarkable facts, the C ¤-algebra is still a bit too general notion to formalize the concepts of noncommutative probability theory. With an extra topological assumption we achieve the desired level of generality. DEFINITION 5 (von Neumann algebra). AC ¤-algebra which is closed not just in the operator norm but also in the weak operator topology is called a von Neumann algebra. Note that the above definition is correct as any C ¤-algebra is isomor- phic to an algebra of bounded linear operators on a Hilbert space H , hence the condition about the closedness in the weak operator topol- ogy makes sense. The weak operator topology on B(H ) is defined by © ª ¯­ ®¯ the family of seminorms p : x, y H where p (A) ¯ Ax, y ¯ (A x,y 2 x,y Æ 2 B(H )). A beautiful result of von Neumann shows that the purely topologi- cal condition of being closed in the weak operator topology can be char- acterized by a purely algebraic condition. In order to present von Neu- mann’s theorem we define the notion of the commutant. DEFINITION 6 (Commutant). Let S be any subset of the algebra B(H ). The commutant of S is denoted by S0 and consists of bounded linear oper- ators on H which commute with each element of S. That is, S0 {B B(H ): AB B A for any A S}. Æ 2 Æ 2 The bicommutant S00 of the set S is defined as the commutant of its commutant.