Coherent Gibbs States: Theory & Applications

• Coherent Gibbs states are defined as statistical ensembles that combine the equilibrium structure of a Gibbs state with the preservation of quantum or classical coherence.
• They use Bayesian model selection and interpolation to reliably estimate state parameters even with finite, noisy data.
• Their applications span quantum thermodynamics, state tomography, and non-Hermitian systems, offering insights into equilibrium behavior and coherent dynamics.

Coherent Gibbs states constitute a broad conceptual class of statistical states that integrate the structural coherence of quantum (or classical) properties with the equilibrium features of the Gibbs ensemble. Their significance extends from quantum statistical inference in small quantum systems, to large-scale thermodynamic models, group field theory and quantum gravity, optical platforms, non-Hermitian quantum mechanics, projected Hilbert spaces, and nonlinear classical and quantum settings. The following provides an authoritative synthesis of the key principles, mathematical structures, methodological advances, and implications across diverse contexts, focusing on their rigorous definition, construction, and role in theory and inference.

1. Formal Definition and Core Properties

Coherent Gibbs states are identified as Gibbs-type ensembles that preserve—by construction or inference—quantum or classical coherence, both in the structure of the state and its evolution or inference under data and constraints. The most general form in finite (or countable) quantum systems is given by

$$\rho = Z^{-1} \exp\left(-\sum_\alpha \lambda_\alpha G_\alpha\right)$$

where $\{G_\alpha\}$ is a specified set of “relevant observables,” and $\{\lambda_\alpha\}$ are Lagrange multipliers determined by constraints such as measured expectation values or data (Rau, 2010).

Coherence, in this context, refers to the retention of off-diagonal elements or phase relations in the chosen basis, as well as invariance or covariance properties under the relevant dynamical maps or measurement updates. A key condition shown in inference settings is that when the prior bias is itself a generalized Gibbs state, Bayesian updating with new data retains the state within the Gibbs manifold—ensuring a form of parametric “coherence” (Rau, 2010, Rau, 2014).

In open quantum systems, coherent Gibbs states arise naturally as non-diagonal steady states in the basis of the “bare” system Hamiltonian due to non-negligible coupling with the environment (“quantum mean force Gibbs states”) (Cresser et al., 2021). In integrable systems, generalized Gibbs ensembles encode entire hierarchies of conserved charges while maintaining integrability-induced coherence in observables (Bastianello et al., 17 Mar 2025).

2. Bayesian Inference and Model Selection

In the context of small quantum systems and incomplete data, the estimation of a coherent Gibbs state involves two Bayesian techniques:

a) Model Selection: Competing sets of relevant observables define different parametrizations of the Gibbs manifold. Bayesian model selection compares marginal likelihoods for each model, balancing data fit with an Occam penalty proportional to the number of parameters. This defines a rule of thumb: additional parameters (i.e. more detailed models) are accepted only if the gain in fit per added parameter exceeds $\ln N$ in likelihood units (Rau, 2010).

b) Bayesian Interpolation: For a given model, the posterior parameters are found via interpolation between prior and data-driven (likelihood) information. Explicitly, for parameter $\lambda$: $$\lambda = \frac{\alpha}{\alpha + N} \lambda_0 + \frac{N}{\alpha + N} \lambda_M$$ where $\lambda_0$ is the prior value, $\lambda_M$ is the maximum likelihood estimate from data, $N$ is the sample size, and $\alpha$ quantifies “confidence” in the prior (Rau, 2010). This smoothly interpolates between prior-dominated and data-dominated estimation, ensuring the resulting state retains the Gibbs form and thus coherence in the relevant subspace.

The combination of the above ensures that, even in low data or ambiguous scenarios, the output state embodies both the equilibrium structure and the coherent features determined by the chosen inferences—subject to explicit error estimates.

3. Quantum Thermodynamic Inference from Tomographic Data

Coherent Gibbs states gain further justification via quantum state tomography, where one reconstructs the state (possibly with incomplete measurements) and tests whether the data are both necessary and sufficient for a consistent Gibbs description. The relevance hypothesis posits that only a specified set of observables determine both the estimate and the update of the quantum state (Rau, 2014).

By demanding that Bayesian updating with either full tomographic data or just the relevant expectation values produce the same posterior, one arrives at a likelihood structure proportional to $\exp[-N S(\mu^\rho_f \| \rho)]$ (with $S$ the quantum relative entropy and $\mu^\rho_f$ the constrained minimizer), directly leading to a Gibbs form for posteriors. Statistical hypothesis testing is performed by comparing deviations from this structure across different sets of observables. If all relevant experimental data can be explained by the Gibbs ansatz (relative to the chosen observables), the resulting states are “coherent” in that they do not require extension beyond the Gibbs manifold (Rau, 2014).

Relevant formulas include:

• Gibbs state: $\mu_g \propto \exp(-\lambda^a G_a)$
• Asymptotic likelihood: $$\mathrm{prob}(\{f_b\} | \rho^{\otimes N}) \sim \exp[-N S(\mu_f^\rho \| \rho)]$$

The approach is notable for its avoidance of theoretical idealizations, focusing solely on finite sequences of experimental measurements and finite exchangeable setups.

4. Mathematical Structure and Generalizations

Several mathematical frameworks are developed across contexts:

Context	State/Operator Formulation	Technical Devices
General quantum system	$$\rho = Z^{-1} \exp(-\sum_\alpha \lambda_\alpha G_\alpha)$$	Bayesian model selection/interpolation (Rau, 2010)
Quantum tomography	$\mu_g \propto \exp(-\lambda^a G_a)$, constrained minimizer	Quantum Sanov, likelihood ∝ $\exp(-N S(\cdot))$
Open quantum systems	Mean force Gibbs state: $\rho_S = \mathrm{Tr}_R(e^{-\beta H_{SR}})/Z$	Weak & ultrastrong coupling expansions (Cresser et al., 2021)
Systems with symmetry	Generalized (Souriau) Gibbs: $\rho(\beta) = Z^{-1} \exp(-\beta \cdot J)$	Momentum map constraints (Chirco et al., 2019)
Nonlinear PDE	Gibbs measure: $\mu_\beta(du) = Z_\beta^{-1} \exp(-\beta h(u)) du$	KMS, Malliavin calculus (Ammari et al., 2021)
Projected Hilbert space	$\rho_P = \mathcal{P} \rho \mathcal{P}$, with $\mathcal{P}$ a projector	CPS basis, P-representation, dynamical equations

Technical advances include the use of relative entropy as a unifying metric (likelihood, hypothesis testing, variational characterization of equilibrium), coherent state representations (for phase‐space and de Finetti limits), and entropy-based reasoning for both quantum and classical cases.

5. Applications in Small and Large Quantum Systems

The framework and formalism of coherent Gibbs states find application in:

• Small quantum systems (single/multiple qubits): Model selection between Ising and Heisenberg descriptions is guided by Bayesian criteria for relevance, with estimation uncertainty reflecting the finite data scenario (Rau, 2010).
• Classical systems: The method applies equally to classical inference problems, such as Jaynes’ analysis of biased dice, using Gibbs formulations and model selection.
• Non-standard settings: Generalizations exist for biorthogonal (pseudo-Hermitian or PT-symmetric) settings, where coherence is maintained under algebraic dynamics and generalized KMS-like conditions, even for non-Hermitian Hamiltonians, via carefully constructed time evolutions (Bagarello et al., 2016).

6. Quantum Coherence, Error Bars, and Physical Implications

A critical aspect of coherent Gibbs states is the retention of physical coherence and proper quantification of uncertainty:

• If the prior is a generalized Gibbs state, all Bayesian updates remain within the manifold, ensuring physical coherence (including off-diagonal elements, phase relationships, and consistent symmetries).
• The estimation method provides explicit error bars, reflecting the statistical uncertainty due to finite sampling, essential in small quantum systems where quantum fluctuations and decoherence cannot be neglected.
• Real-world applications (e.g., incomplete quantum-state tomography) depend on Bayesian corrections to ensure that physically relevant quantum coherence is not lost, as would happen with crude classical probability assignments or overfitting.

7. Broader Implications and Extensions

The analytic and operational foundation outlined for coherent Gibbs states has extended significance across more general domains:

• Group field theory and quantum gravity: The statistical equilibrium procedures based on the maximum entropy principle and KMS flows enable the construction of condensate-like, coherent equilibrium states out of structurally non-spatiotemporal degrees of freedom (Kotecha et al., 2018, Chirco et al., 2019).
• Nonequilibrium and integrable systems: In integrable quantum and classical systems, generalized Gibbs ensembles carry coherent structures inherited from integrability, enabling relaxation to generalized steady states and connecting quantum and classical statistical mechanics (Bastianello et al., 17 Mar 2025).
• Nonlinear PDEs: The unique correspondence between Gibbs measures and KMS equilibrium states for nonlinear Hamiltonian PDEs demonstrates a coherent statistical structure that unifies probabilistic (Gibbs) and dynamical (KMS) perspectives (Ammari et al., 2021).
• Projective and restricted settings: The use of projected coherent state bases, phase‐space representations, and projection operators enables efficient descriptions of systems subject to symmetry or measurement-induced constraints, with direct relevance to coherent Gibbs states in such subspaces (Drummond et al., 2016).
• Open quantum systems: The realization of mean force Gibbs states under conditions of strong system–environment coupling leads to equilibrium states exhibiting nontrivial coherence patterns, with practical impact in nanoscale thermodynamics (Cresser et al., 2021).

Conclusion

Coherent Gibbs states are a unifying concept at the intersection of equilibrium statistical mechanics, quantum inference, and operational quantum theory. They arise whenever the set of relevant observables, prior information, and data constraints are combined by statistical or quantum inference to yield a state that both satisfies the required equilibrium properties and retains coherence under the appropriate definition (algebraic, dynamical, or information-theoretic). Advanced techniques—Bayesian inference, hypothesis testing, relative entropy minimization, phase-space methods, and algebraic approaches—increasingly enable their robust construction even in small, noisy, or otherwise challenging regimes. The continued analysis and refinement of coherent Gibbs states is central to ongoing developments in quantum tomography, open systems, integrable models, condensed matter, and the statistical foundations of quantum gravity.