Gibbs Variational Formula for Thermal Equilibrium States in Terms of Quantum Relative Entropy Density
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Gibbs Variational Formula for Thermal Equilibrium States in Terms of Quantum Relative Entropy Density Hajime Moriya1 Received: 11 March 2020 / Accepted: 19 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Föllmer for classical systems. Keywords Gibbs variational principle · Quantum Gibbs states · Quantum relative entropy Mathematics Subject Classification 82B10 · 82B20 · 46L55
1 Introduction Translation invariant thermal equilibrium states are identified with those attaining minimum free energy. This wisdom of statistical physics is called the Gibbs variational principle, and its rigorous mathematical formulation has been established for classical lattice systems [1] and quantum lattice systems [2]. The Gibbs variational principle can be expressed in terms of the relative entropy (Kullback–Leibler divergence [3]) as stated in [4] for classical systems. In this note, we establish an analogous statement for quantum lattice systems by extending the previous work [5] to a larger class of translation covariant potentials Φ. In more detail, we will prove that the information rate h(ω Φ, β) of any translation invariant state ω with respect to the potential Φ is equal to the relative entropy density h(ω | ψ) of ω with respect to any translation invariant thermal equilibrium state ψ for Φ. This equivalence immediately yields the complete characterization of translation invariant thermal equilibrium states ϕ by the equality condition h(ϕ | ψ) = 0. In [5] under the rather restricted setup that admits only a unique thermal equilibrium state ψ, the equality h(ω Φ, β) = h(ω | ψ) is verified. We prove this equality for a more general case that can have multiple thermal phases (induced by symmetry breaking). In the above
Communicated by Eric A. Carlen.
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Hajime Moriya [email protected] College of Science and Engineering, Kanazawa University, Kanazawa, Japan
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H. Moriya
sense, this paper makes a progress by relating quantum statistical physics to information theory. However, in terms of mathematics, we simply follow the argument invented by HiaiPetz [5] using a recent finding by Ejima-Ogata [6].
2 Preliminaries In this section we give our formulation that is based on C*-algebraic quantum statistical physics [7,8]. We will include a short review of some relevant ideas and known facts for the readers who are not familiar with them.
2.1 Quantum Relative Entropy First we recall the quantum relative entropy by Umegaki [9]. It is a fundamental quantity of this paper. Consider any finite dimensional full matrix algebra Mn (C) (n ∈ N). Let Tr denote the matrix trace which takes 1 on each one-dimensional projection. Let ψ1 and ψ2 be states on Mn (C) whose density matrices with respect to Tr are denoted by D(ψ1 ) and D(ψ2 ).
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