Abstract
Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel L, which give the first order (linear) term of the minimum output p-Rényi entropies of L ⊗r as functions of r. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the p-Rényi entropy for all p≥30.95.
Highlights
Inspired by Montanaro [46], we introduce the concept of additivity rates of a quantum channel L, which give the first order term of the minimum output p-Renyi entropies of L⊗r as functions of r
We show that there exist positive partial transpose (PPT) channels which violate additivity of Renyi p entropy
We investigate how the lower bound for additivity rates behaves with respect to tensor products
Summary
We focus on three questions related to additivity properties of quantum channels. For natural class of random quantum channels, we bound additivity violation of Renyi entropy. The paper is divided roughly into two parts: Sections 3 and 4 deal with the general theory of additivity rates and their lower bounds, while Sections 5-10 deal with random quantum channels.
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