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arXiv · 2105.00462

Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel

Salman Beigi
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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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Salman Beigi. 2021-05-02. Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel. https://doi.org/10.1063/5.0056388

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