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arXiv:0907.3210 · Open Systems & Information Dynamics

On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

Fernando G. S. L. Brandão, Michał Horodecki

Published Mar 1, 2010• 60 citations

Abstract

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.

DOI: 10.1142/s1230161210000047

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•Jan 1, 2014•15 citations

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.

We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-

p

entropy for any

p\in[1,\infty]

. Additionally, given any

p_0>0

and

\mathfrak g>0

, we extend the construction to obtain an explicit non-random channel which has entropy gap

\ge\mathfrak g

uniformly over

p\in[p_0,\infty]

.

Counterexamples to additivity of minimum output

p

-Rényi entropy of quantum channels for all

p\ge 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.

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of ε-net argument and Lévy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm-like structures - positive homogeneity and triangle inequality. Then, a connection between Hastings' method and ours is also discussed. Besides, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels.