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arXiv:0910.1189 · Journal of Mathematical Physics

Nonadditivity of Rényi entropy and Dvoretzky’s theorem

Guillaume Aubrun, Stanisław J. Szarek, Elisabeth Werner

Published Feb 1, 2010• 26 citations
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Abstract

The goal of this note is to show that the analysis of the minimum output p-Rényi entropy of a typical quantum channel essentially amounts to applying Milman’s version of Dvoretzky’s theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output p-Rényi entropy (for p>1).

DOI
10.1063/1.3271044

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The goal of this note is to show that the analysis of the minimum output p-Rényi entropy of a typical quantum channel essentially amounts to applying Milman’s version of Dvoretzky’s theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output p-Rényi entropy (for p>1).
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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.
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-ppp entropy for any p∈[1,∞]p\in[1,\infty]p∈[1,∞]. Additionally, given any p0>0p_0>0p0​>0 and g>0\mathfrak g>0g>0, we extend the construction to obtain an explicit non-random channel which has entropy gap ≥g\ge\mathfrak g≥g uniformly over p∈[p0,∞]p\in[p_0,\infty]p∈[p0​,∞].
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.