Can I read Interpolation of Operators With Trace Inequalities Related To The Positive Weighted Geometric Mean on EtoBox?

Interpolation of Operators With Trace Inequalities Related To The Positive Weighted Geometric Mean by Chayes, Victoria is a scholarly article available to read on EtoBox.

What is Interpolation of Operators With Trace Inequalities Related To The Positive Weighted Geometric Mean about?

There are various generalizations of the geometric mean $(a,b)\mapsto a^{1/2}b^{1/2}$ for $a,b\in \mathbb{R}^+$ to positive matrices, and we consider the standard positive geometric mean $(X,Y)\mapsto X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2}$. Much research in recent years has been devoted to relating the weighted version of this mean $X\#_{t}Y:=X^{1/2}(X^{-1/2}YX^{-1/2})^{t}X^{1/2}$ for $t\in [0, 1]$ with operators $e^{(1-t)X+tY}$ and $e^{(1-t)X/2}e^{tY}e^{(1-t)X/2}$ in Golden-Thompson-like inequalities. These inequalities are of interest to mathematical physicists for their relationship to quantum entropy, relative quantum entropy, and R\'{e}nyi divergences. However, the weighted mean is well-defined for the full range of $t\in\mathbb{R}$. In this paper we examine the value of $|||e^H\#_te^K|||$ and variations thereof in comparison to $|||e^{(1-t)H+tK}|||$ and $|||e^{(1-t)H}e^{tK}|||$ for any unitarily invariant norm $|||\cdot|||$ and in particular the trace norm, creating for the first time the full picture of interpolation of the weighted geometric mean with the Golden-Thompson Inequality. We expand inequalities known for $|||(e^{rH}\#_te^{rK})^{1/r}|||$ with $r>0$, $t\in [0,1]$

Author

Chayes, Victoria

Published

2019

Language

EN