Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, II: Effros’ Proof

AbstractIn 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.

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Monotonicity with volume of entropy and of mean entropy for translationally invariant systems as consequences of strong subadditivity

Journal of Physics A General Physics ◽

10.1088/0305-4470/34/3/304 ◽

2001 ◽

Vol 34 (3) ◽

pp. 365-382 ◽

Cited By ~ 5

Author(s):

Amanda R Kay ◽

Bernard S Kay

Keyword(s):

Strong Subadditivity

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THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

International Journal of Modern Physics D ◽

10.1142/s0218271813420303 ◽

2013 ◽

Vol 22 (12) ◽

pp. 1342030 ◽

Cited By ~ 9

Author(s):

KYRIAKOS PAPADODIMAS ◽

SUVRAT RAJU

Keyword(s):

Hilbert Space ◽

Black Hole ◽

Quantum Mechanics ◽

Nonperturbative Effects ◽

Von Neumann Entropy ◽

Von Neumann ◽

Information Theoretic ◽

Black Hole Evaporation ◽

Strong Subadditivity ◽

Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

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BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

Reviews in Mathematical Physics ◽

10.1142/s0129055x94000407 ◽

1994 ◽

Vol 06 (05a) ◽

pp. 1147-1161 ◽

Cited By ~ 97

Author(s):

MARY BETH RUSKAI

Keyword(s):

Relative Entropy ◽

Discrete Case ◽

Sobolev Inequalities ◽

Completely Positive ◽

Logarithmic Sobolev Inequalities ◽

Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

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On quantum quasi-relative entropy

Reviews in Mathematical Physics ◽

10.1142/s0129055x19500223 ◽

2019 ◽

Vol 31 (07) ◽

pp. 1950022

Author(s):

Anna Vershynina

Keyword(s):

Relative Entropy ◽

Logarithmic Function ◽

Operator Inequalities ◽

Operator Convex Function ◽

Strong Subadditivity ◽

Entropy Formula ◽

Quantum Relative Entropy ◽

Skew Information ◽

Explicit Bounds ◽

Joint Convexity

We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.

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