Hypercontractivity for Semigroups of Unital Qubit Channels

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-t1H1⊗⋯⊗e-tnHn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e^{-t_1 H_1}\otimes \cdots \otimes e^{- t_n H_n}}$$\end{document} to be a contraction from Lp to Lq, where Lp is the algebra of 2n-dimensional matrices equipped with the normalized Schatten norm, and each generator Hj 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.