Hypercontractivity for semigroups of unital qubit channels

Abstract

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form e^{- t_1 H_1} \ot ... \ot e^{- t_n H_n} to be a contraction from L^p to L^q, where L^p is the algebra of 2^n-dimensional matrices equipped with the normalized Schatten norm, and each generator H_j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.