Central Limit Theorem for Tensor Products of Free Variables via Bi-Free Independence

Abstract: In this paper, a connection between bi-free probability and the asymptotics of random quantum channels and tensor products of random matrices is established. Using bi-free matrix models, it is demonstrated that the spectral distribution of certain self-adjoint quantum channels and tensor products of random matrices tend to a distribution that can be obtained by an averaged sum of products of bi-freely independent pairs. Subsequently, using bi-free techniques, a Central Limit Theorem for such operator is established.