Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel

TL;DR

This paper improves the quantum hypercontractivity inequality for the qubit depolarizing channel by establishing a tighter logarithmic-Sobolev inequality, leading to new bounds and applications in quantum information theory.

Contribution

The authors derive an improved quantum logarithmic-Sobolev inequality and leverage it to enhance the hypercontractivity inequality for the qubit depolarizing channel, with applications in quantum inequalities.

Findings

Tighter quantum hypercontractivity inequality established.

Derived an asymptotically optimal quantum Faber-Krahn inequality.

Introduced a new quantum Schwartz-Zippel lemma.

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.