arXiv:2609.20753

Quantum Entropy Contraction and Factorization from Hypercontractivity

Li Gao, Lijun Wang

quant-phmath.FAmath.OASubmitted Sep 17, 2026• Announced Sep 18, 2026• Revised Oct 5, 2026 (v2)

Abstract

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 $σ$, we obtain the modified Log-Sobolev bound $α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}$ where $λ$ is the spectral gap. This removes the assumption of $L_p$-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.

Comments: 15 pages; comments are welcome

Related papers

Complete entropy contraction from conditional hyperboundedness

Li Gao, Lijun Wang

quant-pharXiv:2610.35863Oct 27, 20260 citations

Tight entropy contraction beyond detailed balance

Li Gao, Jingyu Guo

quant-pharXiv:2609.24953Sep 22, 20260 citations

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

Yangjing Dong, Li Gao, Fengning Ou +2 more

quant-pharXiv:2606.17729Jun 17, 20260 citations

Entropy Contraction and Hypercontractivity for Gaussian Quantum Markov Semigroups

Zhengwei Liu, Jincheng Wan, Jinsong Wu

arXiv:2610.06201Oct 5, 20260 citations

Spectral methods for entropy contraction coefficients

Li Gao, Cambyse Rouzé

Feb 8, 20213 citations

Tight Entropy Contraction of Generalized Quantum Depolarization

Li Gao, Long Zhao

quant-pharXiv:2608.19685Aug 21, 20260 citations

arXiv:2609.20753

Quantum Entropy Contraction and Factorization from Hypercontractivity

Li Gao, Lijun Wang

quant-phmath.FAmath.OASubmitted Sep 17, 2026• Announced Sep 18, 2026• Revised Oct 5, 2026 (v2)

arXiv HTML PDF

Abstract

α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}

where

λ

is the spectral gap. This removes the assumption of (L_p)-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.

Comments: 15 pages; comments are welcome

Related papers

Complete entropy contraction from conditional hyperboundedness

Li Gao, Lijun Wang

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 $2\to q$ hyperboundedness at time $t_0>0$ combined with a GNS gap $λ>0$ implies a modified logarithmic Sobolev constant $α=\frac{λ(1-2/q)} {12(λt_0+\log M+1-2/q)},$ where $M$ is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.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 $2\to q$ hyperboundedness at time $t_0>0$ combined with a GNS gap $λ>0$ implies a modified logarithmic Sobolev constant \[ α=\frac{λ(1-2/q)} {12(λt_0+\log M+1-2/q)},\] where $M$ is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.

quant-pharXiv:2610.35863Oct 27, 20260 citations

Tight entropy contraction beyond detailed balance

Li Gao, Jingyu Guo

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.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.

quant-pharXiv:2609.24953Sep 22, 20260 citations

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

Yangjing Dong, Li Gao, Fengning Ou +2 more

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive 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. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$ -hypercontractivity estimates for generalized qubit depolarizing channels.We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive 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. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.

quant-pharXiv:2606.17729Jun 17, 20260 citations

Entropy Contraction and Hypercontractivity for Gaussian Quantum Markov Semigroups

Zhengwei Liu, Jincheng Wan, Jinsong Wu

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 $1In 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

arXiv:2610.06201Oct 5, 20260 citations

Spectral methods for entropy contraction coefficients

Li Gao, Cambyse Rouzé

We propose a conceptually simple method for proving various entropic inequalities by means of spectral estimates. Our approach relies on a two-sided estimate of the relative entropy between two quantum states in terms of certain variance-type quantities. As an application of this estimate, we answer three important problems in quantum information theory and dissipative quantum systems that were left open until now: first, we prove the existence of strictly contractive constants for the strong data processing inequality for quantum channels that satisfy a notion of detailed balance. Second, we prove the existence of the complete modified logarithmic Sobolev constant which controls the exponential entropic convergence of a quantum Markov semigroup towards its equilibrium. Third, we prove a new generalization of the strong subadditivity of the entropy for the relative entropy distance to a von Neumann algebra under a specific gap condition. All the three results obtained are independent of the size of the environment.We propose a conceptually simple method for proving various entropic inequalities by means of spectral estimates. Our approach relies on a two-sided estimate of the relative entropy between two quantum states in terms of certain variance-type quantities. As an application of this estimate, we answer three important problems in quantum information theory and dissipative quantum systems that were left open until now: first, we prove the existence of strictly contractive constants for the strong data processing inequality for quantum channels that satisfy a notion of detailed balance. Second, we prove the existence of the complete modified logarithmic Sobolev constant which controls the exponential entropic convergence of a quantum Markov semigroup towards its equilibrium. Third, we prove a new generalization of the strong subadditivity of the entropy for the relative entropy distance to a von Neumann algebra under a specific gap condition. All the three results obtained are independent of the size of the environment.

Feb 8, 20213 citations

Tight Entropy Contraction of Generalized Quantum Depolarization

Li Gao, Long Zhao

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 $f$ -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.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 $f$-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.

quant-pharXiv:2608.19685Aug 21, 20260 citations

α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}

where

λ

2\to q

hyperboundedness at time

t_0>0

combined with a GNS gap

λ>0

implies a modified logarithmic Sobolev constant

α=\frac{λ(1-2/q)} {12(λt_0+\log M+1-2/q)},

where

M

is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.

(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. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp

(q,2)

-hypercontractivity estimates for generalized qubit depolarizing channels.

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

f

-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.