Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups

Abstract:We prove almost tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) property. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)$, approximately 0.96, times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. As an application of the same method, we also obtain sharp $(q,2)$-hypercontractivity estimates for qubit depolarizing channels.

Quantum Physics (quant-ph); Operator Algebras (math.OA)

arXiv:2606.17729 [quant-ph]

(or arXiv:2606.17729v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.17729 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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