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arXiv:0911.2515 · Journal of Physics A Mathematical and Theoretical

Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for allp> 2

Andrzej Grudka, Michał Horodecki, Łukasz Pankowski

Published Oct 4, 2010• 28 citations
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Abstract

We present a constructive example of the violation of the additivity of minimum output Rényi entropy for each p > 2. The example is provided by an antisymmetric subspace of a suitable dimension. We discuss the possibility of extension of the result to go beyond p > 2 and obtain additivity for p = 0 for a class of entanglement breaking channels.

DOI
10.1088/1751-8113/43/42/425304

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We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every ε∈(0,log⁡2)\varepsilon\in(0,\log2)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 log⁡2−ε\log2-\varepsilonlog2−ε for all p∈[0,∞]p\in[0,\infty]p∈[0,∞]. The output dimension can be chosen to be O(ε−3)O(\varepsilon^{-3})O(ε−3) as ε↓0\varepsilon\downarrow0ε↓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.