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.
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.
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.
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) .
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.
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 .
We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every ε∈(0,log2), there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least log2−ε for all p∈[0,∞]. The output dimension can be chosen to be O(ε−3) as ε↓0.
The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity--rank certificate.