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

Abstract

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup \((Φ_t)_{t\ge 0}\) and every tensor power \(n\), we show that the log-Sobolev constant of the product semigroup \(Φ_t^{\otimes n}\) is at least \(2/(3\ln 2)\approx 0.96\) times the log-Sobolev constant of the single-site semigroup \(Φ_t\), independently of \(n\) and the local dimension \(d\). The proof first establishes an 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.