{
  "id": "2203.06136",
  "version": "v1",
  "published": "2022-03-11T18:09:13.000Z",
  "updated": "2022-03-11T18:09:13.000Z",
  "title": "A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality",
  "authors": [
    "Eric A. Carlen",
    "Elliott H. Lieb"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-11T18:09:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B62",
      "46N50"
    ],
    "keywords": [
      "golden-thompson inequality",
      "monotonicity property",
      "golden-thompson trace inequality",
      "g-t monotonicity theorem",
      "g-t inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}