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arXiv:1509.04689 · Journal of Mathematical Physics

Random matrix techniques in quantum information theory

Benoı̂t Collins, Ion Nechita

Published Dec 10, 2015• 110 citations
arXivHTMLPDF

Abstract

The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.

DOI
10.1063/1.4936880

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The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.
The main emphasis of this work is the mathematical theory of quantum channels and their entropic and information characteristics. Quantum information theory is one of the key research areas, since it leads the way to vastly increased computing speeds by using quantum systems to store and process information. Quantum cryptography allows for secure communication of classified information. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. The past years were marked with impressive progress made by several researchers in solution of some difficult problems, in particular, the additivity of the entropy characteristics of quantum channels. This suggests a need for a book that not only introduces the basic concepts of quantum information theory, but also presents in detail some of the latest achievements.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.
This thesis studies four topics in quantum information theory using tools from representation theory and (high-dimensional) probability theory.First, we study the nonadditivity of minimum output von Neumann and Rényi entropy of quantum channels.A sketch of the proof by Aubrun, Szarek and Werner for nonadditivity of minimum output entropy is presented, and a slight simplification is given.We show that asymptotically H(•) Shannon entropy, or von Neumann entropy D(ρ, σ) Trace distance between two quantum states ρ, σ F(ρ, σ) Fidelity between two quantum states ρ, σ
One of the major achievements of the recently emerged quantum information theory is the introduction and thorough investigation of the notion of a quantum channel which is a basic building block of any data-transmitting or data-processing system. This development resulted in an elaborated structural theory and was accompanied by the discovery of a whole spectrum of entropic quantities, notably the channel capacities, characterizing information-processing performance of the channels. This paper gives a survey of the main properties of quantum channels and of their entropic characterization, with a variety of examples for finite-dimensional quantum systems. We also touch upon the 'continuous-variables' case, which provides an arena for quantum Gaussian systems. Most of the practical realizations of quantum information processing were implemented in such systems, in particular based on principles of quantum optics. Several important entropic quantities are introduced and used to describe the basic channel capacity formulae. The remarkable role of specific quantum correlations-entanglement-as a novel communication resource is stressed.
We show how Free Probability Theory-and in particular, recent results regarding the norm convergence of random matrices-help to better understand im- portant problems in Quantum Information Theory, such as the problem of Minimum Output Entropy additivity.This is a report on joint papers and projects with Ion Nechita, Serban Belinschi, Camille Male and Motohisa Fukuda 1. ENTANGLEMENT AND ADDITIVITY 1.1.Entanglement and entropy.In Quantum Information Theory, a vector subspace CCC of a tensor product A⊗BA\otimes BA⊗B of two Hilbert spaces is said to be entangled iff 000 is the only vector of CCC that can be written as a⊗ba\otimes ba⊗b with a∈A,a\in A,a∈A, b∈Bb\in Bb∈B .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For x∈A⊗Bx\in A\otimes Bx∈A⊗B , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers λ1(x)≥…≥λk(x)≥0\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0λ1​(x)≥…≥λk​(x)≥0 such that x=∑i=1kλiei(x)⊗fi(x)x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)x=∑i=1k​λi​​ei​(x)⊗fi​(x) Here, k=min(k=min(k=min(dim(A), dim(B));ei(x),dim(B));e_{i}(x),dim(B));ei​(x), fi(x)f_{i}(x)fi​(x) are orthonormal vectors.The sequence λ(x)=(λ1(x),…)∈Rk\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}λ(x)=(λ1​(x),…)∈Rk is uniquely defined.Let KC={λ(x),x∈C,∣∣x∣∣=1}⊂RkK_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}KC​={λ(x),x∈C,∣∣x∣∣=1}⊂Rk .This compact set is a subset of the set △k+={y∈R+k:y1≥y2≥⋯≥yk≥0,∑i=1kyi=1}.\triangle_{k}^{+}=\{y\in \mathbb{R}_{+}^{k}:y_{1}\geq y_{2}\geq\cdots\geq y_{k}\geq 0, \sum_{i=1}^{k}y_{i}=1\}.△k+​={y∈R+k​:y1​≥y2​≥⋯≥yk​≥0,∑i=1k​yi​=1}. Besides, we have Δk+⊂Δk\Delta_{k}^{+}\subset\Delta_{k}Δk+​⊂Δk​ , where Δk={x∈R+k∣∑i=1kxi=1}\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}Δk​={x∈R+k​∣∑i=1k​xi​=1} is the (k−1)−(k-1)-(k−1)− dimensional probability simplex.We will also make an abuse of notation and also denote by KCK_{C}KC​ the subset of △k\triangle_{k}△k​ that is obtained by symmetrizing KCK_{C}KC​ under permuting the coordinates.For a positive real number p>0p>0p>0, we recall that the R\'enyi entropy of order ppp of a probability vector x∈Δkx\in\Delta_{k}x∈Δk​ to be $H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification.46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87) .