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 $V_n\otimes W$, 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 $H_n$ and $V_n$ tending to infinity at the same speed in a paper of Belinschi, Collins and Nechita. 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 to provide the smallest known dimension (184) for the dimension an an ancilla space allowing Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where violation of MOE occurs.