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Sharp reverse entropy inequalities for uniform qubit ensembles

Christoph Hartmann

Published Sep 29, 2026• 0 citations
Publisher

Abstract

This computer-assisted preprint proves a sharp reverse relative entropy inequality L >= beta_m chi for uniform ensembles of individually normalized qubit states, for every m >= 2. Here chi is the Holevo quantity and L is the non-optimized quantum lautum information. The proof combines analytic Bloch-vector and Gram-matrix reductions with four interval certificates covering 855,780 boxes. For three matrices in arbitrary finite dimension, the paper also establishes local nonnegativity near faithful pairwise-commuting equality triples, allowing unequal traces and noncommuting perturbations. Further results include normalization bridges, an angular reduction for three qubits, and certified obstructions to proposed pinching and filtering descent arguments. The unrestricted unequal-trace ensemble inequality and the general equality of complete and ordinary modified logarithmic Sobolev constants are not proved. The accompanying archive contains sources, certificate inputs, verifiers, and reproduced results. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Reverse relative entropy and finite-time contraction for conditional expectations (2026).

DOI
10.5281/zenodo.23035375

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•Oct 17, 2026•1 citation
This computer-assisted preprint proves a sharp reverse relative entropy inequality L >= beta_m chi for uniform ensembles of individually normalized qubit states, for every m >= 2. Here chi is the Holevo quantity and L is the non-optimized quantum lautum information. The proof combines analytic Bloch-vector and Gram-matrix reductions with four interval certificates covering 855,780 boxes. For three matrices in arbitrary finite dimension, the paper also establishes local nonnegativity near faithful pairwise-commuting equality triples, allowing unequal traces and noncommuting perturbations. Further results include normalization bridges, an angular reduction for three qubits, and certified obstructions to proposed pinching and filtering descent arguments. The unrestricted unequal-trace ensemble inequality and the general equality of complete and ordinary modified logarithmic Sobolev constants are not proved. The accompanying archive contains sources, certificate inputs, verifiers, and reproduced results. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Reverse relative entropy and finite-time contraction for conditional expectations (2026).
Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant Ω(1/log⁡n)Ω(1/\log n)Ω(1/logn) for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order O(log⁡n⋅log⁡(n/ε))\mathcal{O}(\log n\cdot \log(n/ε))O(logn⋅log(n/ε)). The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.
This preprint proves reverse relative entropy and finite-time contraction inequalities for trace-preserving conditional expectations on finite-dimensional matrix algebras. A spectral-block reduction for an algebra-valued multiplier and a scalar boundary reduction yield explicit constants gamma_C and Lambda_p(C). Complete bounds follow from complete order bounds. For qubit depolarization, the results give 0.940046944825 <= alpha_c < 0.954706390024 in the convention exp(-2 alpha t), and 0.265671889879 < eta_c < 0.270961852526 at retained weight p=1/2. The contraction coefficient compares each state with its own conditional-expectation equilibrium. The paper also proves a stronger bound for commuting states, with sharpness in specified classes, and derives consequences for products of local channels. The exact complete qubit constants and the unrestricted sharp ensemble inequality are not determined here. The accompanying archive contains LaTeX sources, exact inputs, symbolic and interval verifiers, numerical regression checks, and reproduced outputs. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Sharp reverse entropy inequalities for uniform qubit ensembles (2026).
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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.
If a set of von Neumann subalgebras has a trivial intersection in finite dimension, then the sum of relative entropies of a given density to its projection in each such algebra is larger than a multiple of its relative entropy to its projection in the trivial intersection. This results in a subadditivity of relative entropy with a dimension and algebra-dependent, multiplicative constant. As a primary application, this inequality lets us derive relative entropy decay estimates in the form of modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents.
Correction in version 1.1 (2026-10-18) The Decimal/Fraction interval engine distributed with version 1.0 used context-sensitive Decimal unary negation and absolute value on interval endpoints. These operations can round inward, so the recorded P02G certificate in v1.0 should not be treated as a rigorously outward-rounded enclosure. Version 1.1 repairs those operations using exact copy_negate() and copy_abs(), hardens the second interval implementation's row-bound selection, and regenerates the interval reports, checksums, and reproducibility manifest. The certified rectangle, theorem scope, and all 34 mathematical environments in the manuscript remain unchanged. Repaired P02G and the separately implemented mpmath.iv P02J both pass the original theorem gates. The result remains conditional computer-assisted mathematical evidence, not formal machine verification or external peer review. Previous version (v1.0): https://doi.org/10.5281/zenodo.23438614 This unrefereed research preprint establishes a computer-assisted, quantitatively certified non-identifiability result for fragment-size-averaged Holevo information in an N=4 environment with two equiprobable pure pointer-conditioned branches. For every point in an explicitly specified closed rational rectangle of coordinate area 4 × 10^-14, two realizations of the complete averaged Holevo information curve exist: one with minimum pure-common-rest carrier size two and another with carrier size four. The proof combines a uniform parameterized contraction mapping, directed interval arithmetic with a second interval implementation, and an intermediate-value construction. Reproducibility materials include the certification programs, frozen numerical certificates, SHA-256 checksums, and a standalone 200-state numerical comparison against direct 16-dimensional partial traces. Both interval implementations share the underlying analytic Holevo reduction and do not constitute independent symbolic proofs. The conclusions are restricted to N=4, equiprobable pure branches, fragment-size-averaged Holevo information, and the specified carrier definition. They do not establish non-identifiability for arbitrary N, mixed states, unequal priors, or individual labeled-fragment information. This is an unrefereed preprint, not a formally verified or peer-reviewed mathematical theorem. Comprehensive novelty priority is not claimed. Licensing: The manuscript and original documentation/data are provided under CC BY 4.0 where applicable. Original Python research code is provided under the MIT License. The accompanying reproducibility archive specifies licenses on a per-file basis. AI assistance disclosure: OpenAI ChatGPT assisted with mathematical analysis, Python implementation, numerical and interval verification, manuscript preparation, and quality assurance. Claude contributed critical assessment and adversarial review assistance. Neither AI system is a coauthor or an external scientific referee. The human author, Yoichi Tsujisawa, assumes responsibility for the research and its publication.