scholarly journals On quantum quasi-relative entropy

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.

scholarly journals A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY

2010 ◽  
Vol 22 (09) ◽  
pp. 1099-1121 ◽  
Author(s):  
ANNA JENČOVÁ ◽  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Matrix Inequalities ◽  
Strong Subadditivity ◽  
Special Cases ◽  
Joint Convexity

We consider a generalization of relative entropy derived from the Wigner–Yanase–Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K* Ap K B1-p Lieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.

scholarly journals Some trace inequalities for exponential and logarithmic functions

2019 ◽  
Vol 09 (02) ◽  
pp. 1950008
Author(s):  
Eric A. Carlen ◽  
Elliott H. Lieb
Keyword(s):  
Relative Entropy ◽  
Upper Bound ◽  
Entropy Formula ◽  
Quantum Relative Entropy ◽  
Entropy Functionals ◽  
Positive Matrices ◽  
The Right ◽  
Legendre Transforms ◽  
Logarithmic Functions ◽  
Trace Inequalities

Consider a function [Formula: see text] of pairs of positive matrices with values in the positive matrices such that whenever [Formula: see text] and [Formula: see text] commute [Formula: see text] Our first main result gives conditions on [Formula: see text] such that [Formula: see text] for all [Formula: see text] such that [Formula: see text]. (Note that [Formula: see text] is absent from the right side of the inequality.) We give several examples of functions [Formula: see text] to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables [Formula: see text] instead of just [Formula: see text] alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy [Formula: see text], and two others, the Donald relative entropy [Formula: see text], and the Belavkin–Stasewski relative entropy [Formula: see text]. They are known to satisfy [Formula: see text]. We prove that the Donald relative entropy provides the sharp upper bound, independent of [Formula: see text] on [Formula: see text] in a number of cases in which [Formula: see text] is homogeneous of degree [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text]. We also investigate the Legendre transforms in [Formula: see text] of [Formula: see text] and [Formula: see text], and show how our results for these lead to new refinements of the Golden–Thompson inequality.

Monotonicity of quantum relative entropy and recoverability

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1333-1354 ◽  
Author(s):  
Mario Berta ◽  
Marius Lemm ◽  
Mark M. Wilde
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Remainder Term ◽  
Conditional Entropy ◽  
Quantum Information Theory ◽  
Partial Trace ◽  
Quantum Operations ◽  
Strong Subadditivity ◽  
Joint Convexity

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

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.

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

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
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.

scholarly journals Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 763 ◽  
Author(s):  
Ana Costa ◽  
Roope Uola ◽  
Otfried Gühne
Keyword(s):  
Relative Entropy ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
W States ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Local Measurements ◽  
Action At A Distance ◽  
Quantum Steering ◽  
Joint Convexity

The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

scholarly journals Superadditivity of Quantum Relative Entropy for General States

2018 ◽  
Vol 64 (7) ◽  
pp. 4758-4765 ◽  
Author(s):  
Angela Capel ◽  
Angelo Lucia ◽  
David Perez-Garcia
Keyword(s):  
Relative Entropy ◽  
Quantum Relative Entropy
Sign in / Sign up

Export Citation Format

Share Document