%0 Generic %1 carlen2022trace %A Carlen, Eric A. %A Lieb, Elliott H. %D 2022 %K information quantum %T A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality %U http://arxiv.org/abs/2203.06136 %X The Golden-Thompson trace inequality which states that $Tr\, e^H+K łeq Tr\, e^H e^K$ 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 $H= --\Delta +m$ and $K=$ potential, $Tr\, e^H+(1-u)Ke^uK$ is a monotone increasing function of the parameter $u$ for $0u 1$. Our proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\, X^sY^tX^1-sY^1-t Tr\, XY$ for positive operators X,Y and for $12 s,\,t 1 $ and $s+t 32$. The obvious conjecture that this inequality should hold up to $s+t1$, was proved false by Plevnik. We give a different proof of AHO and also give more counterexamples in the $32, 1$ 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.

@misc{carlen2022trace, 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.}, added-at = {2022-03-14T22:55:43.000+0100}, author = {Carlen, Eric A. and Lieb, Elliott H.}, biburl = {https://www.bibsonomy.org/bibtex/2564e846e6402ccca482f6e1058d310f4/gzhou}, description = {A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality}, interhash = {7084f785f43d5c2437728517415775b7}, intrahash = {564e846e6402ccca482f6e1058d310f4}, keywords = {information quantum}, note = {cite arxiv:2203.06136}, timestamp = {2022-03-14T22:55:43.000+0100}, title = {A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality}, url = {http://arxiv.org/abs/2203.06136}, year = 2022 }