Concentration estimates for random subspaces of a tensor product and application to quantum information theory

Abstract

Given a random subspace H n chosen uniformly in a tensor product of Hilbert spaces V n ⊗ W, we consider the collection K n of all singular values of all norm one elements of H n with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of H n, V n tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of K n was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.