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arXiv:2606.17729

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

Yangjing Dong, Li Gao, Fengning Ou, Penghui Yao, Haigang Zhou

quant-phmath.OASubmitted Jun 16, 2026• Announced Jun 17, 2026• 0 citations• Revised Jun 18, 2026 (v2)
v1 · Jun 16, 2026v2 · Jun 18, 2026v3 · Jul 25, 2026
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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.

Comments
Typos corrected, minor improvements to presentation

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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.
Abstract We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated Lp\mathbb {L}_pLp​ L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated Lp\mathbb {L}_pLp​ L p norms are not uniformly convex for 1≤p≤21\le p \le 21≤p≤2 1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.
We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated Lp\mathbb{L}_pLp​ norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated Lp\mathbb{L}_pLp​ norms are not uniformly convex for 1≤p≤21\leq p \leq 21≤p≤2. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.
We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state σσσ, we obtain the modified Log-Sobolev bound α1≥λ(2+log⁡∥σ−1∥∞)α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}α1​≥(2+log∥σ−1∥∞​)λ​ where λλλ is the spectral gap. This removes the assumption of LpL_pLp​-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.
In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the ppp-logarithmic Sobolev constant vanishes at p=1/2p=1/2p=1/2 and a correlated graph-based example for which the ppp-logarithmic Sobolev constant tends to −∞-\infty−∞ as p→0+p\to 0^+p→0+. These examples obstruct a positive uniform reverse-hypercontractivity rate on (0,1)(0,1)(0,1). By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for 1/2<p≤q<11/2< p\leq q < 11/2<p≤q<1 with explicit sufficient times. For each fixed δ>0δ>0δ>0, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on [δ,1][δ, 1][δ,1] and hence quantum reverse hypercontractivity for δ<p≤q<1δ< p\leq q<1δ<p≤q<1 with fixed 0<δ<1/20<δ<1/20<δ<1/2. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for δ<p≤q<1δ< p\leq q< 1δ<p≤q<1 with fixed 0<δ<10<δ<10<δ<1 for connected graphs when n≥3n \geq 3n≥3.
Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroup's hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative LpL_pLp​ spaces to obtain a general hypercontractive inequality from a particular p→qp \rightarrow qp→q-norm bound. Then, we derive a bound on the 2→42 \rightarrow 42→4-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an NNN-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time poly(log⁡(N)){\rm poly}(\log(N))poly(log(N)) by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.
We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum fff-divergence, and also independently, Bogoliubov--Kubo--Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.