Improved quantum hypercontractivity inequality for the qubit depolarizing channel

Abstract

The hypercontractivity inequality for the qubit depolarizing channel Ψt states that ‖Ψt⊗n(X)‖p≤‖X‖q, provided that p ≥ q > 1 and t≥lnp−1q−1. In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber–Krahn inequality on the hypercube and a new quantum Schwartz–Zippel lemma.