AP-1-1 ADDITIVITY PROBLEMS IN QUANTUM INFORMATION THEORY

Alexander Semenovich Holevo

Charles H. Bennett

Vladimír Majerník

Motohisa Fukuda

Benoı̂t Collins

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.$46L65(46L37,46L54,.46L87)$ .

Motohisa Fukuda