Linear algebraic proof of Wigner theorem and its consequences

2017 ◽  
Vol 67 (2) ◽  
Author(s):  
Jáchym Barvínek ◽  
Jan Hamhalter
Keyword(s):  
Linear Algebra ◽  
Relative Entropy ◽  
Quantum Systems ◽  
Algebraic Proof ◽  
Matrix Algebras ◽  
Jordan Homomorphism ◽  
Quantum Relative Entropy ◽  
Rank One

AbstractWe present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.

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 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 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 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
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