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

• URL: http://arxiv.org/abs/2203.06136v1
• Date: Fri, 11 Mar 2022 18:09:13 GMT
• Title: A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality
• Authors: Eric A. Carlen and Elliott H. Lieb
• Abstract summary: The Golden-Thompson trace inequality $Tr, eH+K leq Tr, eH eK$ has proved to be very useful in quantum statistical mechanics. Here we make this G-T inequality more explicit by proving that for some operators, $H=Delta$ or $H= -sqrt-Delta +m$ and $K=$ potential, $Tr, eH+ (1-u)KeuK$ is a monotone increasing function of the parameter $u$ for $0leq
• Score: 1.5229257192293197
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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