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New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels

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New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels by Jeremy Levick A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Doctor of Philosophy in Mathematics Guelph, Ontario, Canada c Jeremy Levick, June, 2015 ABSTRACT New Methods for the Perfect-Mirsky Conjecture and Private Quantum Channels Jeremy Levick Advisor: University of Guelph, 2015 Dr. Rajesh Pereira The Perfect-Mirsky conjecture states that the set of all possible eigenvalues of an n by n doubly stochastic matrix is the union of certain specific n − 1 regular polygons in the complex plane. It is known to be true for n = 1; 2; 3 and recently a counterexample was found for n = 5. We show that the conjecture is true for n = 4. We also modify the original conjecture for n > 4 in order to accommodate the n=5 counterexample, as well as provide evidence for our new conjecture. An important class of quantum channels are the Pauli channels; channels whose Kraus operators are all tensor products of Pauli matrices. A private quantum channel is a quantum channel that is non-injective for some set of inputs. Private quantum channels generalize the classical notion of the one-time pad to the quantum setting. PQCs are most useful when the set of inputs they privatize is isomorphic as an algebra to an algebra of qubits. We use the theory of conditional expectations to show that in certain cases, the set of algebras privatized by a channel is the set of algebras quasiorthogonal to the range of the channel. Quasiorthogonality is an idea from operator algebras that has begun to find applications elsewhere in quantum information. Using these ideas, we prove that conditional expectation log(n) Pauli channels with commuting Kraus operators can always privatize 2 qubits, where n is the number of Kraus operators. We also use ideas from Fourier analysis on finite groups to give a method for finding privatized algebras general Pauli channels. Using character theory, we extend this analysis beyond Pauli channels, and show how to find private algebras for qutrit and other channels. Finally, using ideas from Lie theory, we show a connection between quasiorthogonality and the Cartan decomposition of a Lie algebra, giving new methods for finding algebras privatized by a channel. We also use the KAK decomposition of a Lie algebra to begin looking at ways to encode algebras so they are privatized. Dedication This thesis is dedicated to my parents and grandparents, particularly the memory of my grandfather Hillel Katzman. Without their love and guidance and support throughout my life, this thesis would not have been possible. iv Acknowledgements I would like to thank my advisor Rajesh Pereira, for his invaluable help and support over the years. It has been wonderful having him as an advisor, a mentor, and a role model. I would also like to thank my co-advisor David Kribs for his support, as well as Pal Fischer, and John Holbrook, and the members of my committee: Bei Zheng, Alan Willms, and especially Chi-Kwong Li, who have also contributed to thesis, and whose suggestions have helped to improve it. Susan McCormick and Carrie Tanti in the Mathematics and Statistics office have always been there to help me navigate administrative matters, and have only ever been extremely helpful. My fellow graduate students made working on this thesis not just tolerable, but fun. My officemates Sarah Plosker and Tyler Jackson, as well as Andrew Skelton, Keith Poore, and especially Bryce Morsky have been wonderful to work with, and great friends. v Contents List of Figures viii 1 Introduction 1 1.1 Outline . .1 1.2 Group Theory . .2 1.2.1 Representations . .4 1.3 Lie Groups and Lie Algebras . .8 2 The Four Dimensional Perfect-Mirsky Problem 10 2.1 Positive Matrices . 10 2.2 Stochastic and Doubly Stochastic Matrices . 11 2.2.1 Doubly Stochastic Matrices and Majorization . 12 2.2.2 Multivariate and Directional Majorization . 13 2.3 The Inverse Eigenvalue Problem for Non-Negative Matrices . 15 2.3.1 Karpelevich and the Region Kn ..................... 16 2.4 The Perfect-Mirsky Conjecture . 18 2.4.1 The Rivard-Mashreghi Counterexample . 20 2.4.2 The Four Dimensional Perfect-Mirsky Conjecture . 21 3 Private Quantum Channels and Matrix Conditional Expectations 27 3.1 Quantum Information . 27 3.1.1 Quantum States, Qubits, and Density Matrices . 27 3.2 Quantum Channels . 29 3.2.1 Completely Positive Trace-Preserving Maps . 29 3.3 Private Quantum Channels . 31 3.4 Algebras and Conditional Expectations . 34 3.5 Quasiorthogonality . 36 vi 3.5.1 Group Theory of Pauli Subalgebras . 38 3.6 Private Pauli Channels . 44 3.7 Conditional Expectations onto Operators Spanned by Group Elements . 48 4 Group Theory, Fourier Analysis, and Lie Theory and Private Quantum Channels 52 4.1 Fourier Analysis and Private Quantum Channels . 52 4.2 Lie Theory and Private Pauli Channels . 57 4.3 Global Cartan and KAK Decompositions . 60 4.4 Cartan Decompositions of su(2n) and Quasiorthogonal Algebras of Paulis . 61 4.4.1 Entanglers and Encoding Private Subalgebras . 63 5 Future Work and Conclusions 68 5.1 Perfect-Mirsky . 68 5.2 Private Quantum Channels . 69 Bibliography 69 vii List of Figures 2.1 The region K4 of all eigenvalues of 4 × 4 stochastic matrices . 19 S5 S6 2.2 The Rivard-Mashreghi point is not in k=1 Πk, but is in k=1 Πk ....... 21 2.3 Close-up of the Rivard-Mashreghi point in K4 ................. 25 2.4 The associated eigenvector is convexly dependent . 26 viii Chapter 1 Introduction 1.1 Outline This thesis is concerned with two different problems, the first from matrix analysis, the sec- ond from the intersection of matrix analysis with quantum information theory. The first problem is that of characterizing the region of all possible eigenvalues of all possible n × n doubly stochastic matrices. A famous conjecture on the matter, the Perfect-Mirsky conjec- ture, is known to be true for n ≤ 3, and false for n = 5. The first part of the thesis proves that the Perfect-Mirsky conjecture is true for n = 4, and formulates a new conjecture to account for the falsity of the Perfect-Mirsky conjecture in the case n = 5. We also discuss some lines of reasoning that support our new conjecture. The second part of the thesis concerns the theory of private quantum channels in quantum information theory. As implied by the name, a private quantum channel is a quantum chan- nel which privatizes some inputs. That is, the channel fails to be injective for certain inputs, and therefore an observer cannot distinguish all possible inputs based only on their out- put. In particular, we discuss the connection between private quantum channels and matrix subalgebras, showing conditions under which subalgebras isomorphic to qubit algebras can be privatized. This section relies on a connection between private channels and conditional expectations, and we also make use of the theory of Fourier analysis on certain finite Abelian groups arising from these conditional expectations. We also use Lie theory to investigate the problem of encoding and decoding qubit algebras to be private for certain channels. This introduction contains some of the necessary mathematical background, particularly the group theory and Lie theory necessary for Chapters 3 and 4. In Chapter 2, we discuss the Perfect-Mirsky conjecture. First we introduce some important 1 ideas from the theory of positive matrices, and discuss the related problem of finding the region of all eigenvalues of all n × n stochastic matrices. We discuss important similari- ties between the stochastic and doubly stochastic cases; we also discuss the counterexample disproving the Perfect-Mirsky conjecture in n = 5, and the relationship between doubly stochastic matrices and majorization. Finally, we combine these ideas to obtain a proof of the Perfect-Mirsky conjecture for n = 4, and we end with a new conjecture to replace the Perfect-Mirsky conjecture, and provide some evidence and heuristic reasoning explaining why this new conjecture might be true. Chapter 3 discusses private quantum channels. After a brief review of the important ideas from quantum information, we explain what a private quantum channel is, and give some examples. We explain the connection between private quantum channels and conditional expectations onto matrix subalgebras, and explain how privacy is related to the operator- algebraic idea of quasiorthogonality. We use these ideas to prove limits on the number of qubits privatized by certain Pauli channels, and show explicit encodings achieving these bounds. Lastly, we look at how group-theoretic properties of the Pauli group play a role in the preceding analysis. We proceed to look further into the relationship between group theory and conditional ex- pectations and privacy. We show how representations of groups with certain properties give rise to conditional expectations, and their associated private subalgebras. We also use tools from the theory of bicharacters, and Fourier analysis on finite Abelian groups to show how to find private subalgebras for channels arising from these groups. Finally in Chapter 4 we turn to the connection between Lie theory and private quantum channels, showing how Cartan decompositions of a Lie algebra relate to quasiorthogonality. We show how the KAK decomposition from Lie theory can be used to encode and decode qubit algebras so as to be private for certain quantum channels. 1.2 Group Theory We develop some of the group theory necessary for the later sections of this thesis. We will usually write the group operation multiplicatively, so g h = gh. Recall that two elements a; b of a group commute if ab = ba.
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