Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg 2133 More information about this series at http://www.springer.com/series/304 Farrukh Mukhamedov • Nasir Ganikhodjaev Quantum Quadratic Operators and Processes 123 Farrukh Mukhamedov Dept. of Comput. & Theor. Sciences International Islamic University Malaysia Kuantan, Malaysia ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-22836-5 DOI 10.1007/978-3-319-22837-2 Nasir Ganikhodjaev Dept. of Comput. & Theor. Sciences International Islamic University Malaysia Kuantan, Malaysia ISSN 1617-9692 (electronic) ISBN 978-3-319-22837-2 (eBook) Library of Congress Control Number: 2015952068 Mathematics Subject Classification (2010): 37A50; 47D07; 37A30; 37A55; 46L53; 81P16; 0G07,60G99, 81S25; 60J28, 81R15, 35Q92, 37N25 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To our families Preface Nonlinear mappings appear throughout mathematics, and their range of applications is immense, including the theory of differential equations, the theory of probability, the theory of dynamical systems, mathematical biology, and statistical physics. Most of the simplest nonlinear operators are quadratic. Even in a one-dimensional setting, the behavior of such operators reveals their complicated structure. If one considers multidimensional analogues of quadratic operators, then the situation becomes more complicated, i.e., the investigation of the dynamical behavior of such operators is very difficult. The history of quadratic stochastic operators and their dynamics can be traced back to Bernstein’s work [18]. The continuous time dynamics of this type of operator was considered by Lotka [134] and Volterra [252]. Quadratic stochastic operators are an important source of analysis in the study of dynamical properties and for modeling in various fields such as mathematical economics, evolutionary biology, population and disease dynamics, and the dynamics of economic and social systems. Unfortunately, up to now, there have been no books devoted to the dynamics of quadratic stochastic operators. This omission in the literature gave us the motivation to write a systematic book about such operators. The general objectives of this book are: (i) to give the first systematic presentation of both analytical and probabilistic techniques used in the study of the dynamics of quadratic stochastic operators and corresponding processes; (ii) to establish a connection between the dynamics of quadratic stochastic operators with the theory of Markov processes; and (iii) to give a systematic introduction to noncommutative or quantum analogues of quadratic stochastic operators and processes. The book addresses the most fundamental questions in the theory of quadratic stochastic operators: dynamics, constructions, regularity, and the connection with stochastic processes. This connection means that the dynamics of such operators can be treated as certain Markov or quadratic processes. This interpretation allows us to use the methods of stochastic processes for a better understanding of the limiting behavior of the dynamics of quadratic operators. vii viii Preface Below we provide an overview of the main topics discussed in this book and explain why they have been selected. The starting point of our book is to introduce a quadratic stochastic operator V W S.X; F/ ! S.X; F/ defined on the set of all probability measures S.X; F/ on .X; F/ and to present some motivations to study such operators. The next step is to define and study stochastic processes that are related to the quadratic stochastic operators in the same way as Markov processes are related to linear transformations. After this, it is natural to develop analytic methods for such processes. The last step is to generalize the theory of quadratic stochastic operators and processes to different algebraic structures, including von Neumann algebras. Such quadratic operators are called quantum quadratic stochastic operators (q.q.s.o.s). In this direction, we study the asymptotic properties of dynamical systems generated by q.q.s.o.s. Moreover, we also investigate Markov and quantum quadratic stochastic processes associated with q.q.s.o.s. An essential feature of our exposition is the first systematic presentation of both the classical and quantum theory of quadratic stochastic operators and processes. We combine analytical and probabilistic tools to get a better insight into the dynamics of both classical and quantum quadratic operators. Moreover, we use several methods from the theory of noncommutative probability, matrix analysis, etc. Now we discuss the structure of the book in more detail. The book is divided into eight chapters; at the end of each chapter, we give some comments and references related to the chapter. The first chapter is an introduction where we collect some models, which can be described by quadratic stochastic operators. Chapter 2 is devoted to quadratic stochastic operators (q.s.o.s) defined on a finite-dimensional simplex. In this chapter, we essentially deal with asymptotical stability (or regularity) condition for such operators. Moreover, we show how the dynamics of q.s.o.s are related to some Markov processes. Some relations between the regularity of a q.s.o. and the corresponding Markov process are investigated. In Chap. 3, we introduce quadratic stochastic processes (q.s.p.s) and give examples of such processes. Note that these quadratic processes naturally arise in the study of certain models with interactions, where interactions are described by quadratic stochastic operators. Furthermore, this chapter contains a construction of nontrivial examples of q.s.p.s. Given a q.s.p., one can associate two kinds of processes, which are called marginal processes. One of them is a Markov process. We prove that marginal processes uniquely define q.s.p.s. The weak ergodicity of q.s.p.s is also studied in terms of the marginal processes. In Chap. 4, we develop analytical methods for q.s.p.s. We follow the lines of Kolmogorov’s [121] paper. Namely, we will derive partial differential equations with delaying argument, for quadratic processes of types A and B, respectively. In the previous chapters, we are considering classical (i.e., commutative) quadratic operators. These operators are defined over commutative algebras. However, such operators do not cover the case of quantum systems. Therefore, in Chap. 5 we introduce a noncommutative analogue of a q.s.o., which is called a quantum quadratic stochastic operator (q.q.s.o.). We show that the set of q.q.s.o.s Preface ix is weakly compact. By means of q.q.s.o.s, one can define a nonlinear operator, which is called a quadratic operator. We also study the asymptotical stability of the dynamics of quadratic operators. Chapter 6 is devoted to quantum quadratic stochastic operators (q.q.s.o.s) acting on the algebra of 2  2 matrices M2 .C/. Positive, trace-preserving maps arise naturally in quantum information theory (see, e.g., [199]) and in other situations where one wishes to restrict attention to a quantum system that should properly be considered a subsystem of a larger system which it interacts with. Therefore, we first describe quadratic operators with a Haar state (invariant with respect to the trace). Then q.q.s.o.s with the Kadison–Schwarz property are characterized. By means of such a description, we provide an example of a positive q.q.s.o., which is not a Kadison–Schwarz operator. On the other hand, this characterization is related to a separability condition, which plays an important role in quantum information [17]. We also examine the stability of the dynamics of quadratic operators associated with q.q.s.o.s given on M2 .C/. In Chap. 7, we investigate a class of q.q.s.o.s defined on the commutative algebra `1 . We define the notion of a Volterra quadratic operator and study its properties. It is proved that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, its extreme points are described. Furthermore, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite-dimensional Volterra operators and prove that any power of this sequence converges in the weak topology. Note that in the finite-dimensional setting such operators have been studied by many authors (see, for example, [74, 252]). In Chap. 8, we define a quantum (noncommutative) analogue of quadratic stochastic processes. In our case, such a process is defined on a von Neumann algebra. In this chapter, we essentially study the ergodic principle for these processes. From a physical point of view, this principle means that for sufficiently large values of time a system described by the process does not depend on the initial state of the system. This book is not intended to contain a complete discussion of the theory of quadratic operators, but primarily relates to the asymptotic stability of such operators and associated processes. Moreover, it reflects the interests of the authors in key aspects of this theory. There are many omitted topics that naturally fit into the purview of quadratic operators. However, we have tried to collect the existing references on quadratic stochastic operators. Some of these are discussed in the separate sections entitled “Comments and References.” This book is suitable as a textbook for an advanced undergraduate/graduate level course or summer school in quantum dynamical systems. It can also be used as a reference book by researchers looking for interesting problems to work on, or useful techniques and discussions of particular problems. It also includes the latest developments in the fields of quadratic dynamical systems, Markov processes, and x Preface quantum stochastic processes. Researchers at all levels are likely to find the book inspiring and useful. Kuantan, Malaysia Kuantan, Malaysia May 2015 Nasir Ganikhodjaev Farrukh Mukhamedov Acknowledgements We want to express our warm thanks to our students (M.S. Saburov, A.F. Embong, N.Z.A. Hamza) and the mathematicians who read drafts and offered useful suggestions and corrections, including L. Accardi, W. Bartoszek, R. Ganikhodzhaev, F. Fidaleo, R.A. Minlos, M. Pulka, U.A. Rozikov, and Ya.G. Sinai. We would also like to manifest our heartfelt gratitude towards our parents (Nabikhodja Ganikhodjaev, Jorahon Ziyautdinova, Maksut Mukhamedov, Munira Mukhamedova). We would like to show our love towards our motivating wives (Mahsuma Usmanova, Shirin Mukhamedova) for their patience during the process of this time-consuming book. Also, our thanks goes to Farzona Mukhamedova for her help in rectifying the flaws in our work. The authors also acknowledge the MOE grants FRGS14-116-0357 and FRGS14-135-0376. Finally, the authors are also grateful to the referees for their useful suggestions, which allowed us to improve the presentation. xi Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 2 Quadratic Stochastic Operators and Their Dynamics .. . . . . . . . . . . . . . . . . . 2.1 Quadratic Stochastic Operators . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 One-dimensional q.s.o.s . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Q.s.o.s and Markov Processes. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Asymptotic Stability of q.s.o.s and Markov Processes . . . . . . . . . . . . . . 2.5 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7 7 15 20 25 29 3 Quadratic Stochastic Processes .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Definition of Quadratic Processes . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Examples of Quadratic Stochastic Processes . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Marginal Markov Processes Related to q.s.p.s . . .. . . . . . . . . . . . . . . . . . . . 3.4 Quadratic Stochastic Operators and Discrete Time q.s.p.s . . . . . . . . . . 3.5 Construction of Quadratic Stochastic Processes .. . . . . . . . . . . . . . . . . . . . 3.6 Weak Ergodicity of Quadratic Stochastic Processes . . . . . . . . . . . . . . . . . 3.7 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 31 31 37 40 46 50 54 58 4 Analytic Methods in the Theory of Quadratic Stochastic Processes . . . 4.1 Quadratic Processes with a Finite Set of States . .. . . . . . . . . . . . . . . . . . . . 4.2 Quadratic Processes with a Continuous Set of States . . . . . . . . . . . . . . . . 4.3 Averaging of Quadratic Stochastic Processes . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 The Set E Is Finite.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 The Set E Is a Continuum .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Diffusion Quadratic Processes . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 61 61 66 75 76 81 82 84 5 Quantum Quadratic Stochastic Operators.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85 5.1 Markov Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85 5.2 Quantum Quadratic Stochastic Operators . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 5.3 Quantum Markov Chains and q.q.s.o.s . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 5.4 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 xiii xiv Contents 6 Quantum Quadratic Stochastic Operators on M2 .C/ . . . . . . . . . . . . . . . . . . . 6.1 Description of Quantum Quadratic Stochastic Operators on M2 .C/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Simple Kadison–Schwarz Type q.q.s.o.s . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Non-Simple Kadison–Schwarz Type q.q.s.o.s . . .. . . . . . . . . . . . . . . . . . . . 6.4 An Example of a Non-Simple q.q.s.o. Which Is Not Kadison–Schwarz.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 The Dynamics of the Quadratic Operator Associated with
" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Stability of the Dynamics of Non-Simple q.q.s.o.s . . . . . . . . . . . . . . . . . . 6.7 Example 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8 Example 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.9 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 7 Infinite-Dimensional Quadratic Operators . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Infinite-Dimensional Quadratic Stochastic Operators . . . . . . . . . . . . . . . 7.2 Volterra Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 The Set of Volterra Operators . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 The Limit Behavior of Volterra Operators.. . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Extension of Finite-Dimensional Volterra Operators . . . . . . . . . . . . . . . . 7.6 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 147 150 154 157 163 170 8 Quantum Quadratic Stochastic Processes and Their Ergodic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Quantum Quadratic Stochastic Processes . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Analytic Methods for q.q.s.p.s . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 The Ergodic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Regularity of q.q.s.p.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Expansion of Quantum Quadratic Stochastic Processes .. . . . . . . . . . . . 8.6 The Connection Between the Fibrewise Markov Process and the Ergodic Principle.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.7 Conjugate Quantum Quadratic Stochastic Processes . . . . . . . . . . . . . . . . 8.8 Quantum Quadratic Stochastic Processes and Related Markov Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8.1 Q.q.s.p.s of Type (A) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8.2 Q.q.s.p.s of Type (B) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9 Tensor Products of q.s.p.s and q.q.s.p.s .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.10 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 109 117 122 126 133 138 142 145 173 173 178 181 188 192 195 201 204 205 208 210 215 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 217 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229