Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel

Abstract

The hypercontractivity inequality for the qubit depolarizing channel ${\Psi }_t$ states that $\Vert {\Psi }_t^{\otimes n}(X){\Vert }_p\le \Vert X{\Vert }_q$ provided that $p\ge q>1$ and $t\ge \ln \sqrt{\frac{p-1}{q-1}}$. 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.

17 pages, added a new quantum Schwartz-Zippel lemma

References in corpus (6)

• Hypercontractivity of quasi-free quantum semigroups
• Entropy Production of Doubly Stochastic Quantum Channels
• On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems
• Quantum reverse hypercontractivity
• A modified logarithmic Sobolev inequality for the Hamming cube and some applications
• Strong Converse for Classical-Quantum Degraded Broadcast Channels