Hypercontractivity for semigroups of unital qubit channels

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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⁻ ᵗ¹ ʰ¹... e⁻ ᵗⁿ ʰⁿ 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.