A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality
Mar 11, 202213 pages
Published in:
- J.Math.Phys. 63 (2022) 6, 062203
- Published: Jun 16, 2022
e-Print:
- 2203.06136 [math-ph]
DOI:
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Abstract: (AIP)
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 H=−√−Δ+m 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 12≤s,t≤1, and s+t≤32. 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 32,1 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.- free energy
- entropy
- statistics: quantum
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