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数理解析研究所講究録

APPLICATIONS OF FREE PROBABILITY TO QUANTUM INFORMATION THEORY (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Benoı̂t Collins

Published Dec 1, 2012• 0 citations
OpenAlex

Abstract

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 $C$ of a tensor product $A\otimes B$ of two Hilbert spaces is said to be entangled iff $0$ is the only vector of $C$ that can be written as $a\otimes b$ with $a\in A,$ $b\in B$ .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For $x\in A\otimes B$ , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers $\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0$ such that $x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)$ Here, $k=min($dim(A), $dim(B));e_{i}(x),$ $f_{i}(x)$ are orthonormal vectors.The sequence $\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}$ is uniquely defined.Let $K_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}$ .This compact set is a subset of the set $\triangle_{k}^{+}=\{y\in \mathbb{R}_{+}^{k}:y_{1}\geq y_{2}\geq\cdots\geq y_{k}\geq 0, \sum_{i=1}^{k}y_{i}=1\}.$ Besides, we have $\Delta_{k}^{+}\subset\Delta_{k}$ , where $\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}$ is the $(k-1)-$ dimensional probability simplex.We will also make an abuse of notation and also denote by $K_{C}$ the subset of $\triangle_{k}$ that is obtained by symmetrizing $K_{C}$ under permuting the coordinates.For a positive real number $p>0$, we recall that the R\'enyi entropy of order $p$ of a probability vector $x\in\Delta_{k}$ to be $H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification.$46L65(46L37,46L54,.46L87)$ .

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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 Hp(x):=11−plog⁡∑i=1kxip.H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.Hp(x):=1−p1​log∑i=1k​xip​.2010 Mathematics Subject Cassification.46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87) .
In scope of CPCPCP -convexity theory for C∗C^{*}C∗ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.
We investigate non-random quantum channels generated from the representation the- ory of orthogonal quantum groups to perform numerical computations for estimates of their minimum output entropies. In doing so, we address the violation of the additiv- ity conjecture for minimum output entropies, previously established through probabilistic proofs. This violation is relevant because given two quantum channels whose minimum output entropies sum to a value greater than the minimum output entropy of their tensor product channel it can be shown that more information can be sent with greater fidelity via the two channels in parallel than by using each separately. This is directly related to the relevance of quantum entanglement in quantum information theory. \nWe numerically compute minimum output entropies for single quantum channels and tensor products of quantum channels with respect to both Von Neumann and Renyi en- tropies. We then proceed to compute entropy estimates for nonplanar channels directly with Temperley-Lieb Category Theory. We conclude by discussing which of our choices for channels are optimal for entangled inputs.
The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used again by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy (MOE) for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the MOE.
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.
Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for all p≥0p\ge 0p≥0
The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order p>1p>1p>1, at the von Neumann point p=1p=1p=1, and for sufficiently small positive ppp, while much of the interval 0<p<10<p<10<p<1 has remained open. In this work, we prove that for every p>0p>0p>0 there exist finite-dimensional quantum channels whose minimum output ppp-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for p>3/4p>3/4p>3/4, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.
Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.