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arXiv:1912.00983 · Journal of Mathematical Physics

Quasi-factorization of Quantum Relative Entropy For Subalgebras with Scalar Intersection

Nicholas LaRacuente

quant-phcs.ITSubmitted Dec 2, 2019• Announced Dec 3, 2019• 10 citations• Revised Sep 21, 2020 (v2)
v1 · Dec 2, 2019v2 · Sep 21, 2020v3 · Mar 9, 2021v4 · Apr 19, 2021v5 · Aug 31, 2021v6 · Jun 18, 2022v7 · Oct 30, 2026
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Abstract

The relative entropy of a quantum density matrix to a subalgebraic restriction, denoted $D(ρ\| \mathcal{E}(ρ))$ for a density $ρ$ and conditional expectation $\mathcal{E}$, appears throughout quantum information. Forms of $D(ρ\| \mathcal{E}(ρ))$ include conditional entropy, mutual information, and a variety of resource measures. For subalgebra restrictions given by commuting conditional expectations, the data processing inequality for relative entropy shows subadditivity of $D(ρ\| \mathcal{E}(ρ))$: such a sum of subalgebra-relative entropies is at least as large as the relative entropy to those subalgebras' intersection. For classical entropy, a property known as quasi-factorization implies that even non-commuting conditional expectations are subadditive up to a multiplicative constant. We prove a multiplicative quasi-factorization inequality for finite-dimensional, quantum or quantum-classical subalgebras with intersection being the complex scalars. Multiplicative quasi-factorization derives relative entropy decay estimates in the form of modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents. Hence quasi-factorization is useful in characterizing decoherence.

Journal reference
J. Math. Phys. 63, 122203 (2022)
DOI
10.1063/5.0053698
Comments
18 pages, 1 figure; adds references and corrects numerical values

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The relative entropy of a quantum density matrix to a subalgebraic restriction, denoted D(ρ∥E(ρ))D(ρ\| \mathcal{E}(ρ))D(ρ∥E(ρ)) for a density ρρρ and conditional expectation E\mathcal{E}E, appears throughout quantum information. Forms of D(ρ∥E(ρ))D(ρ\| \mathcal{E}(ρ))D(ρ∥E(ρ)) include conditional entropy, mutual information, and a variety of resource measures. For subalgebra restrictions given by commuting conditional expectations, the data processing inequality for relative entropy shows subadditivity of D(ρ∥E(ρ))D(ρ\| \mathcal{E}(ρ))D(ρ∥E(ρ)): such a sum of subalgebra-relative entropies is at least as large as the relative entropy to those subalgebras' intersection. For classical entropy, a property known as quasi-factorization implies that even non-commuting conditional expectations are subadditive up to a multiplicative constant. We prove a multiplicative quasi-factorization inequality for finite-dimensional, quantum or quantum-classical subalgebras with intersection being the complex scalars. Multiplicative quasi-factorization derives relative entropy decay estimates in the form of modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents. Hence quasi-factorization is useful in characterizing decoherence.
Abstract In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
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.
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.
Quasi-factorization-type inequalities for the relative entropy have recently proven to be fundamental in modern proofs of modified logarithmic Sobolev inequalities for quantum spin systems. In this paper, we show some results of weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e. upper bounds for the BS-entropy between two bipartite states in terms of the sum of two conditional BS-entropies, up to some multiplicative and additive factors. A more detailed version of this paper is accessible at: https://arxiv.org/pdf/2101.10312.pdf
Abstract The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub- σ -algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels.