The Displacement Functional for Quantum Markov Semigroups: A Cramér-Rao Bound for Cumulative Dissipation

Christian Balfagon

Published Apr 3, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.19396273

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The Displacement Functional for Quantum Markov Semigroups: A Cramér-Rao Bound for Cumulative Dissipation

Christian Balfagon

Published Apr 3, 2026• 0 citations

Publisher

Abstract

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.

DOI: 10.5281/zenodo.19396273

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The displacement functional for quantum Markov semigroups: a mean-residual-life criterion, a sharp classification, and relaxation anisotropy

Christian Balfagon

Reproducibility archive for the paper “The displacement functional for quantum Markov semigroups: a mean-residual-life criterion, a sharp classification, and relaxation anisotropy.” The paper studies, for a primitive quantum Markov semigroup with faithful stationary state, the displacement functional J(ρ₀) = ∫₀^∞ D(Φ_t ρ₀ ‖ σ) dt together with the initial divergence and the entropy-production rate, and asks when the dimensionless ratio 𝓡 = J·Σ/r₀² is at least one — a mean-residual-life inequality. It gives a sharp classification for the χ² and Umegaki relative-entropy divergences, an exact increasing-mean-residual-life criterion, the strong-field relaxation-anisotropy floor, and the mixing-time bounds. This archive contains the LaTeX source and compiled PDF, the figure generators, and the verification code. The self-contained script verification/reproduce_numerics.py reproduces every exact rational identity and high-precision numerical value quoted in the paper, using exact sympy arithmetic and mpmath (finishing with “ALL CHECKS PASSED”); verification/certificates/ contains the directed-rounding interval and sum-of-squares programs supporting the global log-convexity certificate and the strictness results. README.md gives the equation-to-test map, reproduction instructions, and the exact dependency versionsReproducibility archive for the paper “The displacement functional for quantum Markov semigroups: a mean-residual-life criterion, a sharp classification, and relaxation anisotropy.” The paper studies, for a primitive quantum Markov semigroup with faithful stationary state, the displacement functional J(ρ₀) = ∫₀^∞ D(Φ_t ρ₀ ‖ σ) dt together with the initial divergence and the entropy-production rate, and asks when the dimensionless ratio 𝓡 = J·Σ/r₀² is at least one — a mean-residual-life inequality. It gives a sharp classification for the χ² and Umegaki relative-entropy divergences, an exact increasing-mean-residual-life criterion, the strong-field relaxation-anisotropy floor, and the mixing-time bounds. This archive contains the LaTeX source and compiled PDF, the figure generators, and the verification code. The self-contained script verification/reproduce_numerics.py reproduces every exact rational identity and high-precision numerical value quoted in the paper, using exact sympy arithmetic and mpmath (finishing with “ALL CHECKS PASSED”); verification/certificates/ contains the directed-rounding interval and sum-of-squares programs supporting the global log-convexity certificate and the strictness results. README.md gives the equation-to-test map, reproduction instructions, and the exact dependency versions

Jul 14, 20260 citations

Exponential Relative Entropy Decay Along Quantum Markov Semigroups

Melchior Wirth

arXiv:2505.07549May 12, 20250 citations

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}

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

2

-photon dissipation to its asymptotic channel to the uniform topology. For multi-mode systems, a generation theorem in concentrated single-sandwich norms supplies the domain properties required for the regularization argument.