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arxiv: 2606.13251 · v1 · pith:CDRJFO7Fnew · submitted 2026-06-11 · 🪐 quant-ph · math-ph· math.MP

Kubo-Martin-Schwinger conditions for non-Hermitian systems

Chen Lan , Luyao Ma , Hao Yang This is my paper

Pith reviewed 2026-06-27 06:21 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords non-Hermitian Hamiltoniansquasi-HermiticityKMS conditionbiorthogonal eigenbasisthermal statespositivityfinite-dimensional quantum mechanics
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The pith

Positivity of the biorthogonal Gibbs functional holds exactly when the Hamiltonian is quasi-Hermitian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the Kubo-Martin-Schwinger thermal equilibrium condition extends to non-Hermitian Hamiltonians that possess real eigenvalues and complete biorthogonal eigenbases. Its central result states that a functional built by weighting left-right eigenvector matrix elements with Boltzmann factors satisfies positivity on all operators of the form A†A if and only if the Hamiltonian is quasi-Hermitian. The proof obtains the required positive metric directly from the eigenprojectors of this functional via the Riesz representation theorem, without assuming any metric in advance. When quasi-Hermiticity holds, the metric-weighted Gibbs state is shown to obey the full set of analytic KMS boundary conditions.

Core claim

For any diagonalisable H in M_d(C) with real spectrum, the biorthogonal Gibbs functional ω_bi(A) = Z_bi^{-1} ∑_n e^{-β E_n} ⟨φ_n|A|ψ_n⟩ satisfies ω_bi(A†A) ≥ 0 for every A if and only if H is quasi-Hermitian. The metric η is recovered from the eigenprojectors of ω_bi by Riesz representation. Under the quasi-Hermitian assumption the η-weighted state ω_η(A) = Z_η^{-1} Tr[η e^{-β H} A] satisfies all three analytic KMS conditions; this property cannot be deduced from the Hermitian partner h = η^{1/2} H η^{-1/2} because the two states differ whenever [η, h] ≠ 0.

What carries the argument

The biorthogonal Gibbs functional ω_bi constructed from the left and right eigenvectors, whose positivity on A†A supplies a metric-free certificate of quasi-Hermiticity.

If this is right

  • The metric η is constructed directly from the eigenprojectors of ω_bi with no prior choice required.
  • The η-Gibbs state satisfies the three analytic KMS conditions via the Hadamard three-line theorem and Bari’s theorem on Riesz bases.
  • The transported state differs from the Gibbs state of the isospectral Hermitian partner whenever [η, h] ≠ 0, so the KMS property is not inherited from Hermitian theory.
  • The construction fails at exceptional points and when the spectrum is complex.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same positivity test could be examined numerically on small random matrices to check the sharpness of the iff statement.
  • The link to the Fagnola–Umanità quantum detailed-balance condition suggests possible extensions to open non-Hermitian systems.
  • Whether an analogous characterisation survives in infinite-dimensional settings remains open once the finite-dimensional biorthogonal assumption is dropped.

Load-bearing premise

The Hamiltonian must be diagonalisable with real spectrum so that a biorthogonal eigenbasis exists and the functional can be written explicitly.

What would settle it

Exhibit a diagonalisable matrix with real eigenvalues that is not quasi-Hermitian yet still yields a positive ω_bi on all A†A, or conversely a quasi-Hermitian matrix whose ω_bi fails positivity.

read the original abstract

We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a systematic analysis through three complementary routes. Our central result is a thermodynamic characterisation of quasi-Hermiticity: for $H \in M_d(\mathbb{C})$ diagonalisable with real spectrum, the biorthogonal Gibbs functional $\omega_{\rm{bi}}(A) = Z_{\rm{bi}}^{-1} \sum_n e^{-\beta E_n}\langle\phi_n|A|\psi_n\rangle$ satisfies $\omega_{\rm{bi}}(A^\dag A) \geq 0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof constructs the metric $\eta$ directly from the eigenprojectors of $\omega_{\rm{bi}}$ via the Riesz representation theorem, with no prior choice of $\eta$, providing a metric-free certificate of quasi-Hermiticity outside the Mostafazadeh--Scholtz framework. Under the full quasi-Hermitian hypothesis, we prove that the $\eta$-Gibbs state $\omega_\eta(A) = Z_\eta^{-1}\, \rm{Tr}[\eta e^{-\beta H}A]$ satisfies all three analytic KMS conditions, using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The result is non-trivial: the transported state $\hat\omega(X) = \rm{Tr}[e^{-\beta h}X\eta]/Z_\eta$ differs from the Gibbs state of the isospectral Hermitian partner $h = \eta^{1/2}H\eta^{-1/2}$ whenever $[\eta,h]\neq 0$, so the KMS property cannot be deduced from the Hermitian theory by similarity. The gap between this result and the full Haag--Hugenholtz--Winnink $C^*$-algebraic framework is identified. Failure modes at exceptional points and for complex spectra are analysed, and the relation to the Fagnola--Umanit\`a quantum detailed balance condition for open systems is discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to extend the Kubo-Martin-Schwinger (KMS) thermal equilibrium condition to non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems. The central result is a thermodynamic characterisation of quasi-Hermiticity: for diagonalisable H with real spectrum, the biorthogonal Gibbs functional ω_bi satisfies ω_bi(A†A) ≥ 0 for all A if and only if H is quasi-Hermitian. This is proved using the Riesz representation theorem applied to eigenprojectors of ω_bi. Additionally, under the quasi-Hermitian hypothesis, the η-Gibbs state is shown to satisfy the three analytic KMS conditions using the Hadamard three-line theorem and Bari's theorem, with the transported state differing from the Hermitian Gibbs state when [η, h] ≠ 0.

Significance. If the central result were correct, it would offer a metric-free certificate of quasi-Hermiticity and a foundation for defining thermal equilibrium states in non-Hermitian quantum systems that satisfy KMS conditions independently of the Hermitian similarity transformation. The analytic proofs and discussion of the gap to the full C*-algebraic framework, as well as failure modes at exceptional points, add value. The distinction that the KMS property cannot be deduced from the Hermitian theory is noteworthy. However, the soundness of the equivalence is in question.

major comments (2)
  1. [central result] The claimed equivalence that ω_bi(A†A) ≥ 0 for all A if and only if H is quasi-Hermitian, for H diagonalisable with real spectrum (as stated in the abstract and the section on the central result), does not hold. In finite dimensions, any such H is quasi-Hermitian by constructing η from the biorthogonal eigenvectors |ψ_n⟩ and |φ_n⟩ with η|ψ_n⟩ = |φ_n⟩, yielding a positive definite metric satisfying ηH = H†η. Nevertheless, the positivity condition fails when the energy levels differ and β is large, for example with H = [[0, 1], [0, 1]], eigenvalues (0,1), β=10, where the Hermitian part of ρ = Z^{-1} ∑ e^{-β E_n} |ψ_n⟩⟨φ_n| acquires a negative eigenvalue, so that ω_bi(A†A) can be negative. This directly contradicts the equivalence.
  2. [Riesz-representation construction] The construction of the metric η from the eigenprojectors of ω_bi via the Riesz representation theorem (in the section on the central result and the Riesz-representation construction) relies on the positivity of the functional to define a valid inner product, but since positivity does not hold for all quasi-Hermitian H satisfying the initial assumptions, the certificate of quasi-Hermiticity is not generally applicable as claimed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying a substantive issue with the central claim. The provided counterexample shows that the stated equivalence between positivity of the biorthogonal Gibbs functional and quasi-Hermiticity does not hold in general. We will revise the manuscript to correct this overstatement, clarify the scope of the positivity condition, and adjust the discussion of the Riesz-representation construction accordingly. The remaining results on the analytic KMS conditions under the quasi-Hermitian hypothesis remain valid and will be retained with appropriate caveats.

read point-by-point responses

  1. Referee: [central result] The claimed equivalence that ω_bi(A†A) ≥ 0 for all A if and only if H is quasi-Hermitian, for H diagonalisable with real spectrum (as stated in the abstract and the section on the central result), does not hold. In finite dimensions, any such H is quasi-Hermitian by constructing η from the biorthogonal eigenvectors |ψ_n⟩ and |φ_n⟩ with η|ψ_n⟩ = |φ_n⟩, yielding a positive definite metric satisfying ηH = H†η. Nevertheless, the positivity condition fails when the energy levels differ and β is large, for example with H = [[0, 1], [0, 1]], eigenvalues (0,1), β=10, where the Hermitian part of ρ = Z^{-1} ∑ e^{-β E_n} |ψ_n⟩⟨φ_n| acquires a negative eigenvalue, so that ω_bi(A†A) can be negative. This directly contradicts the equivalence.

    Authors: We acknowledge that the counterexample is valid: the given matrix is diagonalisable with real spectrum and hence quasi-Hermitian, yet for sufficiently large β the functional fails to be positive. This demonstrates that the 'if' direction of the claimed equivalence does not hold in general. The 'only if' direction may remain of interest, but the bidirectional statement as written in the abstract and central-result section is incorrect. We will revise both locations to remove the equivalence claim, present positivity of ω_bi as a sufficient (but not necessary) condition for certain properties, and include the referee's counterexample to illustrate the limitation. The thermodynamic characterisation will be restated more modestly as a partial characterisation that applies when the positivity condition is additionally imposed. revision: yes

  2. Referee: [Riesz-representation construction] The construction of the metric η from the eigenprojectors of ω_bi via the Riesz representation theorem (in the section on the central result and the Riesz-representation construction) relies on the positivity of the functional to define a valid inner product, but since positivity does not hold for all quasi-Hermitian H satisfying the initial assumptions, the certificate of quasi-Hermiticity is not generally applicable as claimed.

    Authors: The referee correctly notes that the Riesz-representation argument presupposes positivity of ω_bi to obtain a valid inner product. Because positivity is not automatic for all diagonalisable real-spectrum operators, the construction cannot serve as a general metric-free certificate of quasi-Hermiticity. We will revise the relevant section to state explicitly that the construction yields a metric only when the positivity assumption is satisfied, and we will remove any implication that it characterises quasi-Hermiticity without additional conditions. The discussion of its relation to the Mostafazadeh–Scholtz framework will be updated to reflect this restricted scope. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim derived from external Riesz theorem on explicitly defined functional

full rationale

The paper defines ω_bi directly from the biorthogonal eigenbasis of any diagonalisable H with real spectrum (no pre-existing η). The claimed equivalence to quasi-Hermiticity is obtained by applying the Riesz representation theorem to this functional under the positivity assumption, constructing η from its eigenprojectors. This uses an external theorem and does not reduce the conclusion to the input by definition, self-citation, or renaming. No fitted parameters, no load-bearing self-citations, and the derivation chain remains self-contained against the stated mathematical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claims rest on standard functional-analysis theorems and the assumption of a real-spectrum diagonalisable operator; no free parameters or new postulated entities are introduced.

axioms (3)
  • standard math Riesz representation theorem
    Invoked to construct the metric η directly from the eigenprojectors of ω_bi.
  • standard math Hadamard three-line theorem
    Used to establish the analytic KMS boundary conditions.
  • standard math Bari's theorem on Riesz bases
    Applied to the η-Gibbs state under the quasi-Hermitian hypothesis.

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discussion (0)

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Reference graph

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