Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel

• URL: http://arxiv.org/abs/2105.00462v2
• Date: Thu, 9 Dec 2021 08:08:31 GMT
• Title: Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel
• Authors: Salman Beigi
• Abstract summary: The hypercontractivity inequality for the qubit depolarizing channel $Psi_t$ states that $|Psi_totimes n(X)|_pleq |X|_q$ provided that $pgeq q> 1$ and $tgeq ln sqrtfracp-1q-1$. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev hypercontractivity inequalities to obtain our main result
• Score: 2.9443230571766845
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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