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arXiv:2610.06201

Entropy Contraction and Hypercontractivity for Gaussian Quantum Markov Semigroups

Zhengwei Liu, Jincheng Wan, Jinsong Wu

Published Oct 5, 2026• 0 citations
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Abstract

In this paper, we study finite-mode Gaussian quantum Markov semigroups with a faithful invariant Gaussian state. We present an algebraic characterization of the complete modified logarithmic Sobolev inequality relative to the fixed-point algebra, and we obtain the corresponding optimal constant under the assumption that the drift evolution $e^{t\mathbf{Z}}$ converges as $t\to\infty$. We establish hypercontractivity for such quantum Markov semigroups with Hurwitz drift, in the sense that every fixed pair $1

DOI
10.48550/arxiv.2610.06201

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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.
For finite-dimensional quantum channels compatible with a faithful conditional expectation, we show that conditional hyperboundedness and strict GNS contraction together yield a relative entropy contraction. For KMS-symmetric quantum Markov semigroups, a conditional 2→q2\to q2→q hyperboundedness at time t0>0t_0>0t0​>0 combined with a GNS gap λ>0λ>0λ>0 implies a modified logarithmic Sobolev constant α=λ(1−2/q)12(λt0+log⁡M+1−2/q), α=\frac{λ(1-2/q)} {12(λt_0+\log M+1-2/q)},α=12(λt0​+logM+1−2/q)λ(1−2/q)​, where MMM is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.
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.
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.
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 quantitative entropy contraction bounds for finite-dimensional quantum channels beyond detailed balance. For channels compatible with a faithful conditional expectation, we show that GNS (Gelfand--Naimark--Segal) norm contraction yields relative entropy contraction with only a logarithmic loss in the dimensional constant. For quantum Markov semigroups with a faithful asymptotic conditional expectation, this gives a lower bound on the complete modified logarithmic Sobolev constant in terms of the GNS spectral gap. The bounds allow initial entanglement with an external environment and are independent of its system size. Applications to alternating reservoir pulses and continuously coupled energy ladders give bounds with optimal size dependence.