Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel

Abstract: The hypercontractivity inequality for the qubit depolarizing channel $\Psi_t$ states that $|\Psi_t^{\otimes n}(X)|_p\leq |X|_q$ provided that $p\geq q> 1$ and $t\geq \ln \sqrt{\frac{p-1}{q-1}}$. 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.