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Reverse relative entropy and finite-time contraction for conditional expectations

Christoph Hartmann

Published Sep 29, 2026• 0 citations
Publisher

Abstract

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 Developed with AI assistance; not peer reviewed or formally verified.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).

DOI
10.5281/zenodo.23035262

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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 fff-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.
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).
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.
Contraction coefficients quantify the loss of distinguishability induced by a channel and provide a strong form of the data-processing inequality. While standard contraction coefficients ignore auxiliary quantum systems, existing extensions based on complete contraction coefficients require the compared states to have identical reference marginals. In this work, we introduce conditional contraction coefficients, a novel family that incorporates arbitrary quantum reference systems by subtracting the distinguishability already present in the reference system. We develop a general framework for contraction coefficients with such quantum side information, including the corresponding strong-data-processing-inequality (SDPI) constants, expansion coefficients, and relative contraction coefficients. For the trace distance, we show that the optimization can be restricted to orthogonal input states. For the quantum relative entropy, we prove that its conditional contraction coefficient is exactly equal to the contraction coefficient of the conditional mutual information, extending the classical correspondence between relative-entropy contraction and mutual-information contraction to the setting with quantum side information. More generally, we identify structural properties of divergences required for these results and discuss extensions beyond the relative entropy. These results establish a unified framework for analyzing information contraction in quantum network settings, where quantum side information and distributed correlations are intrinsic features of the information-processing task. Applications include an extension of the Polyanskiy-Wu bounds on mutual information contraction, new perspectives on mixing times, and fundamental limits on quantum memories.
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