A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality

TL;DR

This paper refines the Golden-Thompson inequality by establishing a monotonicity property for certain quantum operators, using a new proof of an inequality by Ando, Hiai, and Okubo, with implications for quantum statistical mechanics.

Contribution

It provides a new proof of the AHO inequality, extends its validity range for quantum operators, and proves a monotonicity property of the Golden-Thompson inequality in quantum mechanics.

Findings

Proved the monotonicity of Tr e^{H+(1-u)K}e^{uK} for specific quantum operators.

Extended the validity of the AHO inequality to a broader parameter range under certain positivity conditions.

Demonstrated the relevance of the inequality to quantum statistical mechanics and free energy comparisons.

Abstract

The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leq Tr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here we make this G-T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, $H=\Delta$ or $H= -\sqrt{-\Delta +m}$ and $K=$ potential, $Tr\, e^{H+(1-u)K}e^{uK}$ is a monotone increasing function of the parameter $u$ for $0\leq u \leq 1$. Our proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\, X^sY^tX^{1-s}Y^{1-t} \leq Tr\, XY$ for positive operators X,Y and for $\tfrac{1}{2} \leq s,\,t \leq 1 $ and $s+t \leq \tfrac{3}{2}$. The obvious conjecture that this inequality should hold up to $s+t\leq 1$, was proved false by Plevnik. We give a different proof of AHO and also give more counterexamples in the $\tfrac{3}{2}, 1$ range. More importantly we show that the inequality conjectured in AHO does indeed hold in this range if $X,Y$ have a certain positivity property -- one which does hold for quantum mechanical operators, thus enabling us to prove our G-T monotonicity theorem.