Approximate tensorization of the relative entropy for noncommuting conditional expectations

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. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as 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.

arXiv comment31 pages. We have modified the structure of the paper, added an Outlook, included a new main result (Theorem 3) and updated our results with some recent advances in the literature

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Cited by in corpus (12)

• Rapid thermalization of spin chain commuting Hamiltonians
• Complete entropic inequalities for quantum Markov chains
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions
• Learning quantum many-body systems from a few copies
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• General Continuity Bounds for Quantum Relative Entropies
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Deviation bounds and concentration inequalities for quantum noises
• Weak quasi-factorization for the Belavkin-Staszewski relative entropy
• Complete Sobolev Type Inequalities
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains