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arXiv:2610.35863

Complete entropy contraction from conditional hyperboundedness

Li Gao, Lijun Wang

quant-phmath.FAmath.OASubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

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.

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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 σσσ, we obtain the modified Log-Sobolev bound α1≥λ(2+log⁡∥σ−1∥∞)α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}α1​≥(2+log∥σ−1∥∞​)λ​ where λλλ is the spectral gap. This removes the assumption of LpL_pLp​-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.
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