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Spectral methods for entropy contraction coefficients

Li Gao, Cambyse Rouzé

Published Feb 8, 2021• 3 citations
OpenAlex

Abstract

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.

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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.
Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L2L_2L2​ L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.
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.
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.
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.
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.
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.