Almost one bit violation for the additivity of the minimum output entropy

Abstract

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that the maximum is attained on a vector of shape (a,b,...,b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any eps > 0, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as log 2 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

v3: appendix replaced by Lemma 3.8

References in corpus (7)

• Hastings' additivity counterexample via Dvoretzky's theorem
• Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0
• Towards a state minimizing the output entropy of a tensor product of random quantum channels
• Revisiting additivity violation of quantum channels
• Asymptotically well-behaved input states do not violate additivity for conjugate pairs of random quantum channels
• Low entropy output states for products of random unitary channels
• Estimates for compression norms and additivity violation in quantum information

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