The Golden–Thompson trace inequality, which states that Tr eH+K ≤ Tr eHeK, 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 = Δ or and K = potential, Tr eH+(1−u)KeuK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr XsYtX1−sY1−t ≤ Tr XY for positive operators X, Y and for , and . The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem.
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June 2022
Research Article|
June 16 2022
A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality
Special Collection:
Special collection in honor of Freeman Dyson
Eric A. Carlen
;
Eric A. Carlen
a)
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
1
Department of Mathematics, Hill Center, Rutgers University
, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
a)Author to whom correspondence should be addressed: [email protected]
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Elliott H. Lieb
Elliott H. Lieb
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
2
Departments of Mathematics and Physics, Jadwin Hall, Princeton University
, Princeton, New Jersey 08544, USA
Search for other works by this author on:
a)Author to whom correspondence should be addressed: [email protected]
Note: This paper is part of the Special Collection in Honor of Freeman Dyson.
J. Math. Phys. 63, 062203 (2022)
Article history
Received:
March 11 2022
Accepted:
May 17 2022
Citation
Eric A. Carlen, Elliott H. Lieb; A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality. J. Math. Phys. 1 June 2022; 63 (6): 062203. https://doi.org/10.1063/5.0091111
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