We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable $H\in M_d(\mathbb C)$ with real spectrum, the biorthogonal Gibbs functional $ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangle$ satisfies $ω_{\rm bi}(A^\dagger A)\ge0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $η$ directly from the eigenprojectors of $ω_{\rm bi}$ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the $η$ -Gibbs state $ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]$ satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever $[η,h]\neq0$ , so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the $η$ -Gibbs state in finite dimensions, while the construction of a compatible $C^*$ -norm and the proof of $σ$ -weak continuity remain open.We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable $H\in M_d(\mathbb C)$ with real spectrum, the biorthogonal Gibbs functional $ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangle$ satisfies $ω_{\rm bi}(A^\dagger A)\ge0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $η$ directly from the eigenprojectors of $ω_{\rm bi}$ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the $η$-Gibbs state $ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]$ satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever $[η,h]\neq0$, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the $η$-Gibbs state in finite dimensions, while the construction of a compatible $C^*$-norm and the proof of $σ$-weak continuity remain open.

quant-pharXiv:2606.13251Jun 12, 20260 citations

Stability, detailed balance and KMS condition for quantum systems

Alberto Frigerio, Vittorio Gorini, Maurizio Verri

Springer eBooksApr 6, 20081 citation

H\in M_d(\mathbb C)

with real spectrum, the biorthogonal Gibbs functional

ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangle

satisfies

ω_{\rm bi}(A^\dagger A)\ge0

for all

A

if and only if

H

is quasi-Hermitian. The proof reconstructs the metric

η

directly from the eigenprojectors of

ω_{\rm bi}

via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the

η

-Gibbs state

ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]

satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever

[η,h]\neq0

, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the

η

-Gibbs state in finite dimensions, while the construction of a compatible

C^*

-norm and the proof of

σ

-weak continuity remain open.