Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.14020

Entropy Maximization and Weak Gibbsianity of Quasi-Free Fermionic States

Vojkan Jakšić, Claude-Alain Pillet, Anna Szczepanek

math-phcond-mat.stat-mechquant-phSubmitted Mar 14, 2026• Announced Mar 17, 2026• Revised Aug 21, 2026 (v2)
v1 · Mar 14, 2026v2 · Aug 21, 2026
arXivHTMLPDF

Abstract

In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function $\widehat C$ satisfies $0<\widehat C(k)<1$ and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.

Journal reference
Lett Math Phys 116, 103 (2026)
DOI
10.1007/s11005-026-02100-9
Comments
13 pages

Related papers

A note on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

Jak\v{s}i\'c, Vojkan, Claude‐Alain Pillet, Clément Tauber

We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice $\mathbb Z^d$. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i\'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

arXiv:2204.02177•Apr 5, 2022•0 citations

Entropy

Robert Alicki, M. Fannes

This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy such as concavity, sub-additivity, strong sub-additivity, and continuity. It proves the existence of mean entropy for shift-invariant states on quantum spin chains, and then derives the expression for the mean entropy for quasi-free Fermions on a chain. The chapter defines the quantum relative entropy and presents its basic properties, including behaviour with respect to completely positive maps. The maximum entropy principle defines thermal equilibrium states (Gibbs states). This variational principle is illustrated by the Hartree–Fock approximation for a model of interacting Fermions. The entropy of an equilibrium state for a free quantum particle on a compact Riemannian manifold is also estimated. Finally, the notion of relative entropy is formulated in the algebraic setting using the relative modular operator.

Oxford University Press eBooks•Jul 12, 2001•0 citations

Equilibrium States and Their Macroscopic Uniformity in Fermion Lattice Systems

Nobuyuki AKIHO

We study equilibrium states and their macroscopic uniformity in the fermion lattice system satisfying the assumptions presented by Araki and Moriya [5]. We show that every translation-invariant KMS state of the even part of the fermion algebra extends uniquely to a translation-invariant KMS state of the whole fermion algebra. The macroscopic uniformity is established for an extremal translation-invariant equilibrium state of the fermion algebra.

Interdisciplinary Information Sciences•Jan 1, 2006•0 citations

Entropy Scaling Law and the Quantum Marginal Problem

Isaac H. Kim

Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.

arXiv:2010.07424•Physical Review X•May 20, 2021•12 citations

Nonfreeness and related functionals for measuring correlation in many-fermion states

Alex D. Gottlieb, Norbert J. Mauser

This article is a brief review of "nonfreeness" and related measures of "correlation" for many-fermion systems. The many-fermion states we deem "uncorrelated" are the gauge-invariant quasi-free states. Uncorrelated states of systems of finitely many fermions we call simply "free" states. Slater determinant states are free; all other free states are "substates" of Slater determinant states or limits of such. The nonfreeness of a many-fermion state equals the minimum of its entropy relative to all free states. Correlation functionals closely related to nonfreeness can be defined in terms of Rényi entropies; nonfreeness is the one that uses Shannon entropy. These correlation functionals all share desirable additivity and monotonicity properties, but nonfreeness has some additional attractive properties.

arXiv:1510.04573•Oct 15, 2015•3 citations

EQUILIBRIUM STATES AND THEIR ENTROPY DENSITIES IN GAUGE-INVARIANT C*-SYSTEMS

Nobuyuki AKIHO, Fumio Hiai, Dėnes Petz

A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.

arXiv:math/0409601•Reviews in Mathematical Physics•May 1, 2005•3 citations
In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function C^\widehat CC satisfies 0<C^(k)<10<\widehat C(k)<10<C(k)<1 and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.
We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Zd\mathbb Z^dZd. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i\'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.
This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy such as concavity, sub-additivity, strong sub-additivity, and continuity. It proves the existence of mean entropy for shift-invariant states on quantum spin chains, and then derives the expression for the mean entropy for quasi-free Fermions on a chain. The chapter defines the quantum relative entropy and presents its basic properties, including behaviour with respect to completely positive maps. The maximum entropy principle defines thermal equilibrium states (Gibbs states). This variational principle is illustrated by the Hartree–Fock approximation for a model of interacting Fermions. The entropy of an equilibrium state for a free quantum particle on a compact Riemannian manifold is also estimated. Finally, the notion of relative entropy is formulated in the algebraic setting using the relative modular operator.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
This article is a brief review of "nonfreeness" and related measures of "correlation" for many-fermion systems. The many-fermion states we deem "uncorrelated" are the gauge-invariant quasi-free states. Uncorrelated states of systems of finitely many fermions we call simply "free" states. Slater determinant states are free; all other free states are "substates" of Slater determinant states or limits of such. The nonfreeness of a many-fermion state equals the minimum of its entropy relative to all free states. Correlation functionals closely related to nonfreeness can be defined in terms of Rényi entropies; nonfreeness is the one that uses Shannon entropy. These correlation functionals all share desirable additivity and monotonicity properties, but nonfreeness has some additional attractive properties.
A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.