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School of Industrial and Information Engineering Master of Science in Mathematical Engineering School of Industrial and Information Engineering Master of Science in Mathematical Engineering RANDOM MATRIX THEORY AND APPLICATIONS IN TELECOMMUNICATION AND QUANTUM SYSTEMS Author : Zheng LI Supervisor : Franco FAGNOLA Academic Year 2018/2019 To C.S. Ying Abstract The study of random matrices started in 1940s, when the physicists observed that the empirical spectral distributions of random Hamiltonians tends to a semicir- cle. Since then, more and more research results about random matrices have been published, and random matrix theory turned out to have deep connections with free probability and combinatorics. In the meantime, random matrix theory has been applied in many other fields, in almost all the situations where one wants to know the asymptotic property of some statistics determined by spectra of large matrices. In this thesis, we present an overview of random matrix theory and illustrate its applications to: telecommunication MIMO systems to the computa- tion of channel capacities, CDMA systems in the evaluation of minimum mean square errors and spectral efficiency, in quantum information for the study of quantum channel capacities and the celebrated conjecture on additivity of quan- tum entropy, in open quantum systems for finding spectra of random Lindblad operators. Keywords: Random Matrix Theory, Telecommunication Systems, Open Quan- tum Systems. iv Acknowledgements Firstly I would like to express my sincere gratitude to Prof. F. Fagnola, who has supervised the whole writing of my thesis. He has given me clear guidelines on studying of different subjects, and was always responsible when I encountered difficulties. I would also thank Prof. V. Moretti, who spent much time to demon- strate some proofs for me with patience. I wish to thank T. Kletti, who kindly read the manuscript and gave me sug- gestions, and always helped me on studying of mathematics. I also wish to thank K. Dong for meaningful discussions on telecommunication systems, thank S.F. Zhang for help on English language writing. I also have to thank my family, in particular my uncle D.S. Li, who is the main financial supporter of my studies here in Italy, and unfortunately passed away last year; I hope he would be glad to know the accomplishment of my master thesis writing in heaven. Finally I want to express a special gratitude to the Department of Mathematics of Politecnico di Milano, which gave me, once a layman, a precious opportunity to study mathematics, and has opened my eyes to see a new beautiful world. vi Contents Abstract iv Acknowledgement vi 1 Introduction1 2 Preliminaries3 2.1 Information Theory............................3 2.1.1 Complex Random Vector....................3 2.1.2 Entropy and Mutual Information................5 2.2 Estimation Theory............................7 2.2.1 Minimal Mean Squared Error Estimator............7 2.2.2 Linear MMSE Estimator.....................7 2.3 Probability Measures on Metric Space.................9 2.3.1 Weak Convergence of Probability Measures..........9 2.3.2 Tightness and Relative Compactness.............. 11 2.3.3 Other Types of Convergence.................. 12 2.4 Bounded Linear Operators on Hilbert Space.............. 15 2.4.1 Banach Algebra and C*-Algebra................ 16 2.4.2 Adjoint Operator......................... 17 2.4.3 Isometry and Partial Isometry.................. 18 2.4.4 Trace Class Operator....................... 21 2.4.5 Von Neumann Algebra..................... 23 3 Random Matrix Theory 27 3.1 Empirical Spectral Distribution..................... 27 3.2 Convergence of Random Distributions................. 28 3.2.1 From Deterministic Distribution to Random Distribution.. 28 3.2.2 General Facts on Convergence of Random Distributions.. 29 3.2.3 Common Types of Convergence Used in RMT........ 30 3.3 Stieltjes Transform............................ 32 3.3.1 Definition and Basic Properties of Stieltjes Transform.... 32 3.3.2 Derivation of semi-circular Law Using Stieltjes Transform. 35 viii 3.4 Asymptotic Results in Random Matrix Theory............ 36 3.4.1 Wigner Matrices and semi-circular Law............ 36 3.4.2 Wishart Matrices and Marchenko-Pastur Distribution.... 37 3.4.3 Ginibre Matrices and Circular Law............... 38 3.4.4 ESD of Another Important Class of Random Matrices.... 39 3.5 Convergence Rates of ESD........................ 40 3.5.1 In Cases of Wigner Matrices and Wishart Matrices...... 40 3.5.2 Simulation of Convergence Rate of ESD............ 42 3.6 Connections with Free Probability................... 43 3.6.1 Non-commutative Probability Space and Freeness...... 43 3.6.2 Free Product and Free Probability............... 45 3.6.3 Free Central Limit Theorem and Asymptotic Freeness.... 47 4 Applications of RMT 49 4.1 In MIMO System............................. 49 4.1.1 Asymptotic Result I: Fixed Number of Receivers....... 50 4.1.2 Asymptotic Result II: Simultaneously Tending to Infinity.. 51 4.2 In CDMA System............................. 53 4.2.1 Cross-correlations of Random Spreading Sequences..... 55 4.2.2 MMSE Multiuser Dectection and Spectral Efficiency..... 56 4.2.3 Other Types of Detections in CDMA System......... 60 4.3 In Open Quantum System........................ 61 4.3.1 Quantum State and Quantum Channel............ 61 4.3.2 Asymptotic Minimal Output Entropy............. 65 4.3.3 Spectrum of Random Quantum Channel........... 68 4.3.4 Quantum Markov Semigroup and Lindblad Equation.... 71 4.3.5 Random Matrix Model of Lindbladian............. 74 5 Conclusions and Future Development 76 Bibliography 78 ix Chapter 1 Introduction Random matrix theory (RMT) first appeared in the study of quantum mechan- ics in the 1940s. In quantum mechanics, values of physical observables such as energy and momentum are regarded as eigenvalues of linear bounded operators on a Hilbert space. In particular, the Hermitian operator Hamiltonian, which is closely related to the time-evolution of a quantum system (this will also be mentioned in our Section 4.3.4), played the vital role in the theory of quantum mechanics. Hence, the asymptotic behavior (in particular the distribution of the spectrum) of large dimensional random matrices of such type had attracted spe- cial interests, and semi-circular law was discovered during that time [2]. Later, lots of researchers started to work on this field. Many other types of matrices have been studied, and the convergence of the empirical spectral distri- butions of random matrices has been proven in more and more strong sense. In 1980s, L. Pastur [5], as a pioneer, introduced the Stieltjes transform into random matrix theory, which can discover the limit spectral distribution in many cases, without knowing prior knowledge. Moreover, Z.D. Bai et al. [11][12] have used the Stieltjes transform as the main tool to study the convergence rate of empirical spectral distributions. It is worth mentioning that, in the meantime, D. Voiculescu created the free probability, a theory that studies non-commutative random vari- ables, in which the limit distribution in the free version of central limit theorem is the semi-circular law. About 1991, D. Voiculescu [9] also discovered that freeness could be asymptotically held for many kinds of random matrices. Nowadays, the random matrix theory has been applied to numerous fields, anywhere we want to know the asymptotic property of some statistics depending on the spectra of matrices. In this master thesis, we first introduce some preliminary knowledge on sev- eral quite different topics in Chapter2, which will be used in the subsequent chapters. Notice that we will select only the knowledge that is out of the cur- riculum of the Mathematical Engineering program at Politecnico di Milano; in other words, we assume that the reader is familiar with the basics in Algebra, Probability, Measure Theory, and Functional Analysis. 1 In Chapter3, we will clarify the formal definition of "convergence" for the ran- dom probability measures, and give some properties of the aforementioned pow- erful tool Stieltjes transform. The limit spectral distributions of different types of ensemble like Wigner ensemble, Wishart ensemble and Ginibre ensemble, will be listed. Moreover, we will discuss the free probability and its connections to random matrix theory. In Chapter4 one can find several applications of random matrix theory in telecommunication systems and open quantum systems. The first application appears in multiple-input multiple-output (MIMO) system, in which the channel can be modelled as a matrix, and the Information Theory tells us the capacity of the channel is determined by the eigenvalues of that matrix, hence we can ap- ply the random matrix theory to analyze the asymptotic capacity of the channel. The second application is about code-division multiple access (CDMA) system, in which people use random spreading sequences to modulate the signal, and the linear minimal mean square error estimation to demodulate signal. We will analyze the asymptotic error and capacity of such estimation. The third appli- cation is about the asymptotic capacity of the random quantum channel, and it depends on the eigenvalues in a random subspace of a tensor product. The last application is about the sampling of the spectrum of random Lindblad opera- tor, in high dimensional open quantum systems. We will give a random matrix model of such operator, which conserves the asymptotic spectral property, but dramatically reduces the sampling time. 2 Chapter 2 Preliminaries We begin with some preliminary but important results that will be used in the
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