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arXiv:0712.3628 · Communications in Mathematical Physics

Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0

Toby S. Cubitt, Aram W. Harrow, Debbie Leung, Ashley Montanaro, Andreas Winter

Published Sep 9, 2008• 43 citations
arXivHTMLPDF
DOI
10.1007/s00220-008-0625-z

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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.
In this paper, we present new families of quantum channels for which corresponding minimum output Rényi p-entropy is not additive. Our manuscript is motivated by the results of Grudka et al. and we focus on channels characterized by both extensions and subspaces of the antisymmetric subspace inCd⊗Cd\mathbb {C}^{d} \otimes \mathbb {C} ^{d}Cd⊗Cd, which exhibit additivity breaking forp>2p\gt 2p>2.