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.

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 Coefficient inequalities for a class of analytic functions associated with the lemniscate of Bernoulli

2018 ◽  
Vol 37 (4) ◽  
pp. 83-95
Author(s):  
Trailokya Panigrahi ◽  
Janusz Sokól
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Toeplitz Determinant ◽  
Second Hankel Determinant ◽  
Coefficient Inequalities ◽  
The Right

In this paper, a new subclass of analytic functions ML_{\lambda}^{*}  associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}|  for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

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

scholarly journals Faithful measure of quantum non-Gaussianity via quantum relative entropy

2019 ◽  
Vol 100 (1) ◽  
Author(s):  
Jiyong Park ◽  
Jaehak Lee ◽  
Kyunghyun Baek ◽  
Se-Wan Ji ◽  
Hyunchul Nha
Keyword(s):  
Relative Entropy ◽  
Quantum Relative Entropy

scholarly journals An Ergodic Theorem for the Quantum Relative Entropy

2004 ◽  
Vol 247 (3) ◽  
pp. 697-712 ◽  
Author(s):  
Igor Bjelakovic ◽  
Rainer Siegmund-Schultze
Keyword(s):  
Ergodic Theorem ◽  
Relative Entropy ◽  
Quantum Relative Entropy

scholarly journals MONOTONICITY OF QUANTUM RELATIVE ENTROPY REVISITED

2003 ◽  
Vol 15 (01) ◽  
pp. 79-91 ◽  
Author(s):  
DÉNES PETZ
Keyword(s):  
Relative Entropy ◽  
Coarse Graining ◽  
Von Neumann Entropy ◽  
Trace Inequality ◽  
Von Neumann ◽  
Quantum Relative Entropy ◽  
Crucial Property

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

scholarly journals A Second Law for Open Markov Processes

2016 ◽  
Vol 23 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Blake S. Pollard
Keyword(s):  
Free Energy ◽  
Markov Process ◽  
Markov Processes ◽  
Relative Entropy ◽  
Upper Bound ◽  
Equilibrium Distribution ◽  
Detailed Balance ◽  
Rate Of Change ◽  
Second Law ◽  
Boundary States

In this paper we define the notion of an open Markov process. An open Markov process is a generalization of an ordinary Markov process in which populations are allowed to flow in and out of the system at certain boundary states. We show that the rate of change of relative entropy in an open Markov process is less than or equal to the flow of relative entropy through its boundary states. This can be viewed as a generalization of the Second Law for open Markov processes. In the case of a Markov process whose equilibrium obeys detailed balance, this inequality puts an upper bound on the rate of change of the free energy for any non-equilibrium distribution.

scholarly journals Monotonicity of the Quantum Relative Entropy Under Positive Maps

2017 ◽  
Vol 18 (5) ◽  
pp. 1777-1788 ◽  
Author(s):  
Alexander Müller-Hermes ◽  
David Reeb
Keyword(s):  
Relative Entropy ◽  
Quantum Relative Entropy ◽  
Positive Maps
Sign in / Sign up

Export Citation Format

Share Document