Interpolation of Operators With Trace Inequalities Related To The Positive Weighted Geometric Mean

Abstract

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 $r>0$, $t\in [0,1]$ to the entire real line, and comment on how the exterior inequalities can be used to provide elegant proofs of the known inequalities for $t\in [0,1]$. We also characterize the equality cases for strictly increasing unitarily invariant norms.