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arXiv:2203.03745 · Quantum

Self-restricting Noise and Exponential Relative Entropy Decay Under Unital Quantum Markov Semigroups

Nicholas D. LaRacuente

Published Mar 4, 2026• 1 citation

Abstract

States of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.

DOI: 10.22331/q-2026-03-04-2010

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States of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.

We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.

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.

This paper introduces the displacement functional

\mathcal{J}(\rho_{0}) := \int_{0}^{\infty} S(\Phi_{t}(\rho_{0})||\sigma) dt

for quantum Markov semigroups (QMS) with a unique faithful fixed point

\sigma

. The functional measures the cumulative relative-entropy dissipation along the trajectory from an initial state

\rho_{0}

to the equilibrium state

\sigma

. Main Results: Quantum Cramér-Rao Inequality: We prove a fundamental lower bound for

\mathcal{J}

given by

\mathcal{J}(\rho_{0}) \ge S(\rho_{0}||\sigma)^{2} / EP_{\mathcal{L}}(\rho_{0})

, where

EP

is the entropy production rate. The "Sandwich" Bound: Combined with the Modified Logarithmic Sobolev Inequality (MLSI), we establish that the total dissipation is bounded both above and below:

\frac{S}{2\Lambda} \le \mathcal{J} \le \frac{S}{2\alpha}

. Rigidity Theorem: We demonstrate that the lower bound is sharp and is saturated if and only if the system's entropy decay follows a single exponential trajectory. Structural Stability: We prove that these properties are stable under perturbations of the generator. This work provides a missing piece in the study of quantum entropic convergence, offering a non-perturbative lower bound that complements existing upper bounds in open quantum 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.

In this work, we extend the theory of quantum Markov processes on a single quantum state to a broader theory that covers Markovian evolution of an ensemble of quantum states, which generalizes Lindblad’s formulation of quantum dynamical semigroups. Our results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Φ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium. In particular, we study the convergence rates of two special semigroups, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Φ-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity—the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Φ-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. There is no classical analogy for these different equilibrium situations. Finally, we also study a statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes and provide upper bounds to the mixing time.