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 ${e^{-t_1 H_1}otimes cdots otimes e^{- t_n H_n}}$ 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.