Luận án tiến sĩ: Các hàm khoảng cách trong lý thuyết thông tin lượng tử và vấn đề liên quan - Vương Trung Dũng

MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY VUONG TRUNG DUNG SOME DISTANCE FUNCTIONS IN QUANTUM INFORMATION THEORY AND RELATED PROBLEMS DOCTORAL DISSERTATION IN MATHEMATICS BINH DINH – 2024 MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY VUONG TRUNG DUNG SOME DISTANCE FUNCTIONS IN QUANTUM INFORMATION THEORY AND RELATED PROBLEMS Speciality: Mathematical Analysis Speciality code: 9 46 01 02 Reviewer 1: Prof. Dang Duc Trong Reviewer 2: Prof. Pham Tien Son Reviewer 3: Assoc. Pham Quy Muoi Supervisors: 1.

Le Cong Trinh 2. Dinh Trung Hoa BINH DINH – 2024 Declaration This thesis was completed at the Department of Mathematics and Statistics, Quy Nhon Uni- versity under the supervision of Assoc. Le Cong Trinh and Assoc. Dinh Trung Hoa.

I hereby declare that the results presented in it are new and original. Most of them were published in peer-reviewed journals, others have not been published elsewhere. For using results from joint papers I have gotten permission from my co-authors. Binh Dinh, 2024 Vuong Trung Dung i Acknowledgments This thesis was undertaken during my years as a PhD student at the Department of Math- ematics and Statistics, Quy Nhon University.

Upon the completion of this thesis, I am deeply indebted to numerous individuals. On this occasion, I would like to extend my sincere appreci- ation to all of them. First and foremost, I would like to express my sincerest gratitude to Assoc. Dinh Trung Hoa, who guided me into the realm of matrix analysis and taught me right from the early days.

Not only that, but he also devoted a significant amount of valuable time to engage in dis- cussions, and provided problems for me to solve. He motivated me to participate in workshops and establish connections with senior researchers in the field. He guided me to find enjoyment in solving mathematical problems and consistently nurtured my enthusiasm for my work. I can’t envision having a more exceptional advisor and mentor than him.

The second person I would like to express my gratitude to is Assoc. Le Cong Trinh, who has been teaching me since my undergraduate days and also introduced me to Prof. From the early days of sitting in lecture halls at university, Prof. Trinh has been instilling inspiration and a love for mathematics in me.

It’s fortunate that now I have the opportunity to be mentored by him once again. He has always provided enthusiastic support not only in my work but also in life. Without that dedicated support, it would have been difficult for me to complete this thesis. I would like to extend a special thank you to the educators at both the Department of Math- ii ematics and Statistic and the Department of Graduate Training at Quy Nhon University for providing the optimal environment for a postgraduate student who comes from a distant loca- tion like myself.

Binh Dinh is also my hometown and the place where I have spent all my time from high school to university. The privilege and personal happiness of coming back to Quy Nhon University for advanced studies cannot be overstated. I am grateful to the Board and Colleagues of VNU-HCM High School for the Gifted for providing me with much support to complete my PhD study. Especially, I would like to extend my heartfelt gratitude to Dr.

Nguyen Thanh Hung, who has assisted to me in both material and spiritual aspects since the very first days I set foot in Saigon. He is not only a mentor and colleague but also a second father to me, who not only supported me financially and emotionally during challenging times but also constantly encouraged me to pursue a doctoral degree. Without this immense support and encouragement, I wouldn’t be where I am today. I also want to express my gratitude to Su for the wonderful time we’ve spent together, which has been a driving force for me to complete the PhD program and strive for even greater achieve- ments that I have yet to attain.

Lastly, and most significantly, I would like to express my gratitude to my family. They have always been by my side throughout work, studies, and life. I want to thank my parents for giving birth to me and nurturing me to adulthood. This thesis is a gift I dedicate to them.

Binh Dinh, 2024 Vuong Trung Dung iii Contents Declaration i Acknowledgment ii Glossary of notation vi Introduction 1 1 Preliminaries 13 1.1 Matrix theory fundamentals .2 Matrix function and matrix mean. 19 2 Weighted Hellinger distance 28 2.1 Weighted Hellinger distance .2 In-betweenness property. 32 3 The α-z-Bures Wasserstein divergence 38 3.1 The α-z-Bures Wasserstein divergence and the least squares problem .2 Data processing inequality and in-betweenness property .3 Quantum fidelity and its parameterized versions .4 The α-z-fidelity between unitary orbits. 75 4 A new weighted spectral geometric mean 82 4.1 A new weighted spectral geometric mean and its basic properties .2 The Lie-Trotter formula and weak log-majorization.

87 Bibliography 103 v Glossary of notation Cn : The set of all n-tuples of complex numbers 〈x, y〉 : The scalar product of vectors x and y Mn : The set of n × n complex matrices H : The Hilbert space B(H) : The set of all bounded linear operators acting on Hilbert space H Hn : The set of all n × n Hermitian matrices H+ n : The set of all n × n positive semi-definite matrices Pn : The set of all n × n positive definite matrices I, O : The identity and zero elements of Mn , respectively A∗ : The conjugate transpose (or adjoint) of the matrix A |A| : The positive semi-definite matrix (A∗ A)1/2 Tr(A) : The canonical trace of matrix A λ(A) : The vector of eigenvalues of matrix A in decreasing order s(A) : The vector of singular values of matrix A in decreasing order Sp(A) : The spectrum of matrix A 󰀂A󰀂 : The operator norm of matrix A |||A||| : The unitarily invariant norm of matrix A x≺y : x is majorized by y x ≺w y : x is weakly majorized by y A󰂒B : The geometric mean of two matrices A and B vi A󰂒t B : The weighted geometric mean of two matrices A and B A󰂑B : The spectral geometric mean of two matrices A and B A󰂑t B : The weighted spectral geometric mean of two matrices A and B Ft (A, B) : The F -mean of two matrices A and B A∇B : The arithmetic mean of two matrices A and B A!B : The harmonic mean of two matrices A and B A:B : The parallel sum of two matrices A and B µp (A, B, t) : The matrix p-power mean of matrices A and B vii Introduction Quantum information stands at the confluence of quantum mechanics and information the- ory, wielding the mathematical elegance of both realms to delve into the profound nature of information processing at the quantum level. In classical information theory, bits are the funda- mental units representing 0 and 1. Quantum information theory, however, introduces the concept of qubits, the quantum counterparts to classical bits. Unlike classical bits, qubits can exist in a superposition of states, allowing them to be both 0 and 1 simultaneously.

This unique property empowers quantum computers to perform certain calculations exponentially faster than classical computers. Entanglement is a crucial phenomenon in quantum theory where two or more particles be- come closely connected. When particles are entangled, changing the state of one immediately affects the state of the other, no matter the distance between them. This has important impli- cations for quantum information and computing, offering new possibilities for unique ways of handling information.

Quantum algorithms, such as Shor’s algorithm for factoring large numbers and Grover’s algorithm for quantum search, exemplify the power of quantum information in tackling complex computational tasks with unparalleled efficiency. In order to treat information processing in quantum systems, it is necessary to mathemati- cally formulate fundamental concepts such as quantum systems, states, and measurements, etc. Useful tools for researching quantum information are functional analysis and matrix theory. First, we consider the quantum system.

It is described by a Hilbert space H, which is called a representation space. This will be advantageous because it is not only the underlying basis of 1 quantum mechanics but is also as helpful in introducing the special notation used for quantum mechanics. The (pure) physical states of the system correspond to unit vectors of the Hilbert space. This correspondence is not 1-1.

When f1 and f2 are unit vectors, then the correspond- ing states are identical if f1 = zf2 for a complex number z of modulus 1. Such z is often called phase. The pure physical state of the system determines a corresponding state vector up to a phase. Traditional quantum mechanics distinguishes between pure states and mixed states.

Mixed states are described by density matrices. A density matrix or statistical operator is a pos- itive matrix of trace 1 on the Hilbert space. This means that the space has a basis consisting of eigenvectors of the statistical operator and the sum of eigenvalues is 1. In quantum information theory, distance functions are used to measure the distance between two mixed states.

Addition- ally, these distance functions can be employed to characterize the properties of a given quantum state. For instance, they can quantify the quantum entanglement between two parts of a state, representing the shortest distance between the state and the set of all separable states. These distance functions naturally extend to the set of positive semi-definite matrices, which is also the main focus of this thesis. Nowadays, the significance of matrix theory has been widely recognized across various fields, including engineering, probability and statistics, quantum information, numerical analy- sis, biological and social sciences.

In image processing (subdivision schemes), medical imaging (MRI), radar signal processing, statistical biology (DNA/genome), and machine learning, data from numerous experiments are stored as positive definite matrices. To work with each set of data, we need to select its representative element. In other words, we need to compute the aver- age of the corresponding positive definite matrices. Therefore, considering global solutions of the least-squares problems for matrices is of paramount importance (refer to [2, 8, 18, 28, 67, 73] for examples).

Consider the following least squares problem: d2 (x, a) + d2 (x, b) → min, x ∈ [a, b], 2 where d := dE (x, y) = |y − x|, or, d := dR (x, y) := | log(y) − log(x)|. √ The arithmetic mean (a + b)/2 and the geometric mean ab are unique solutions to the above problem with respect to dE and dR distance, respectively. Moreover, based on the AM-GM inequality for two non-negative numbers a and b, we have a new distance as follows a+b √ d(a, b) = − ab. 2 For A, B ∈ Pn , some matrix analogs of scalar distances are: • Euclidean distance induced from Euclidean/Frobenius inner product 〈A, B〉 = Tr(A∗ B).

The associated norm is 󰀂A󰀂F = 〈A, A〉1/2 = (Tr(A∗ A))1/2. 󰀣 n 󰀤1/2 󰁛 • The Riemann distance [12] is δR (A, B) = || log(A−1 B)||2 = log2 λi (A−1 B). i=1 • Bures-Wasserstein distance [13] in the theory of optimal transport : 󰀓 󰀓󰀃 󰀄 󰀔󰀔1/2 1/2 1/2 1/2 db (A, B) = Tr(A + B) − 2 Tr A BA. • The Log-Determinant metric [75] in machine learning and quantum information: A+B dl (A, B) = log det − 2 log det(AB).

2 • The Hellinger metric or Bhattacharya metric [73] in quantum information : 󰀃 󰀃 󰀄󰀄1/2 dh (A, B) = Tr(A + B) − 2 Tr A1/2 B 1/2. In applications, one are sometimes interested in distance-like functions that provide distance 3 between two data points.