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arXiv:1603.00577

Haagerup's inequality and additivity violation of the Minimum Output Entropy

Benoı̂t Collins

Published Mar 2, 2016• 6 citations
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Abstract

We give a simple and conceptual proof of the fact that random unitary channels yield violation of the Minimum Output Entropy additivity. The proof relies on strong convergence of random unitary matrices and Haagerup's inequality.
DOI
10.48550/arxiv.1603.00577

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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.
Non-Additivity of Minimum Output p-Reˊnyi\mathbf{R\acute{e}nyi}Reˊnyi Entropy
Hastings disproved additivity conjecture for minimum output entropy by using random unitary channels. In this note, we employ his approach to show that minimum output p−p-p−Rényi entropy is non-additive for p∈(0,p0)∪(1−p0,1)p\in(0,p_0)\cup(1-p_0,1)p∈(0,p0​)∪(1−p0​,1) where p0≈0.2855p_0\approx 0.2855p0​≈0.2855.
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.