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 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.
ABSTRACT. The free contraction norm (or the (t)-norm) was introduced in [BCN12] 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 in [BCN13] in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. This free con-traction 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 minimum output entropy. 1.
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⊗B of two Hilbert spaces is said to be entangled iff 0 is the only vector of C that can be written as a⊗b with a∈A,b∈B .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For x∈A⊗B , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers λ1(x)≥…≥λk(x)≥0 such that x=∑i=1kλiei(x)⊗fi(x) Here, k=min(dim(A), dim(B));ei(x),fi(x) are orthonormal vectors.The sequence λ(x)=(λ1(x),…)∈Rk is uniquely defined.Let 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}. Besides, we have Δk+⊂Δk , where Δk={x∈R+k∣∑i=1kxi=1} is the (k−1)− dimensional probability simplex.We will also make an abuse of notation and also denote by KC the subset of △k that is obtained by symmetrizing KC 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∈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) .
Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel L , which give the first order (linear) term of the minimum output p -Rényi entropies of L ⊗ r as functions of r . We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the p -Rényi entropy for all p ≥ 30 . 95 .
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.
Counterexamples to additivity of minimum output p-Rényi entropy of quantum channels for all p≥0
The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order p>1, at the von Neumann point p=1, and for sufficiently small positive p, while much of the interval 0<p<1 has remained open. In this work, we prove that for every p>0 there exist finite-dimensional quantum channels whose minimum output p-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for p>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.
The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by Matthew B. Hastings [“Superadditivity of communication capacity using entangled inputs”, Nature Physics 5, 255–257 (2009; doi:)] in the one-shot case by exhibiting a pair of random quantum channels. However, the initial motivation was arguably to understand regularized quantities, and there was so far no way to solve additivity questions in the regularized case. The purpose of this paper is to give a solution to this problem. Specifically, we exhibit a pair of quantum channels that unearths additivity violation of the regularized minimum output entropy. Unlike previously known results in the one-shot case, our construction is non-random, infinite-dimensional, and in the commuting-operator setup. The commuting-operator setup is equivalent to the tensor-product setup in the finite-dimensional case for this problem, but their difference in the infinite-dimensional setting has attracted substantial attention and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's and Connes embedding problem [ Z. Ji et al., “ \mathsf{MIP}^*= \mathsf{RE} ”, Preprint, ], Likewise, it is not clear that our approach works in the finite-dimensional setup. Our strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.