Researchers are increasingly investigating how quantum coherence impacts thermodynamic systems. Maurizio Fagotti from Université Paris-Saclay, CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, and colleagues demonstrate a novel framework for understanding energy fluctuations that retain genuinely coherent contributions. Their work introduces a method for ‘foliating’ state space into minimum-variance leaves, determined by the Fisher information, and constructing leaf-canonical ensembles. This collaborative research, involving analysis across theoretical and statistical modelling, establishes a structure that extends beyond equilibrium, proposing a ‘leaf typicality’ hypothesis where local observables depend solely on leaf and energy levels. This advancement offers a significant step towards a more complete understanding of eigenstate thermalization and the role of coherence in complex quantum systems.
Scientists have developed a novel thermodynamic framework allowing for genuinely quantum-coherent contributions to energy fluctuations, challenging the conventional view of equilibration as a late-time phenomenon. This work introduces a method for organizing quantum states based on “minimum-variance leaves”, geometrical regions defined by minimising energy variance across all possible pure-state decompositions.
The research demonstrates that traditional statistical ensembles, such as the Gibbs ensemble, emerge as special cases within this broader structure, existing on a specific, commuting leaf. By foliating state space in this manner, researchers establish a foundation for extending the concept of eigenstate thermalization, the idea that individual energy eigenstates resemble thermal ensembles, beyond equilibrium conditions.
The core of this advancement lies in the construction of a “leaf-canonical ensemble”, a statistical description of states that prioritizes minimal coherent energy fluctuations. This approach moves beyond the traditional reliance on stationary states, instead focusing on identifying large equivalence classes of locally indistinguishable states. Crucially, the minimal average energy variance is directly linked to the quantum Fisher information, a measure of the sensitivity of a quantum state to changes in energy.
This connection reveals that the optimal decomposition of states within a leaf remains consistent even when considering convex combinations of states, simplifying the analysis and providing a robust mathematical foundation. Researchers define an “effective Hamiltonian”, termed Hρ, which governs the behaviour of states within each leaf and reduces to the standard Hamiltonian when the state commutes with it.
The populations of states within a leaf are then determined by the eigenvectors and eigenvalues of Hρ, offering a unique and precise way to characterise the ensemble. This formulation allows for the construction of a nonequilibrium generalisation of thermodynamic entropy, maximising the Shannon entropy of populations subject to normalization and fixed mean energy.
The resulting leaf-canonical ensemble provides a powerful tool for understanding the dynamics of many-body quantum systems and potentially predicting their long-term behaviour. Data supporting the leaf-typicality hypothesis were gathered for the complete set of local observables spanning one or two neighboring sites within the minimum-variance ensemble.
These calculations utilised a non-integrable Hamiltonian with parameters set to (√5+5/8, √5/2, π/20), differing from the transverse-field Ising point described previously. The research explored states thermal with respect to H0, employing an inverse temperature β of 0.25, while H0 maintained the same functional form with parameters (0, 1/2, 0). Chain lengths of L = log2 d were examined, specifically at 6, 8, 10, and 12, with increasing line thickness in the figures denoting larger system sizes.
Typicality diagnostics, measuring the convergence towards an eigenstate thermalization hypothesis-like result, were calculated for each observable. At β = 0.25, the energy incoherence reached approximately 0.97 log d, demonstrating a strong alignment with the leaf-typicality prediction. Further analysis at β = 0.75 revealed an energy incoherence of roughly 0.76 log d, indicating a slight slowing of convergence.
A significantly reduced energy incoherence of 0.24 log d was observed at β = 1.75, suggesting that convergence becomes progressively slower as incoherence decreases. To demonstrate the limits of the hypothesis, the roles of H and H0 were reversed, preparing a thermal state of H and examining the foliation induced by the integrable Hamiltonian H0. The resulting typicality diagnostic failed to sharpen with increasing system size, consistent with the expectation that additional conserved quantities prevent the observed convergence.
Specifically, the energy incoherence in this reversed scenario was approximately 0.92 log d. These results collectively demonstrate the dependence of leaf typicality on the underlying Hamiltonian and the degree of energy coherence within the system. A central methodological component of this work involves foliating state space into “minimum-variance leaves”, a technique designed to isolate and characterise coherent contributions to energy fluctuations.
This process begins by defining leaves based on minimising the average energy variance across all possible pure-state decompositions, with the minimum variance dictated by the Fisher information, a measure of the amount of information that an observable carries about an unknown parameter. Constructing these leaves allows for the creation of a “leaf-canonical ensemble”, the least-biased statistical state consistent with normalization and mean energy constraints for each leaf.
The Gibbs ensemble, a standard result in statistical mechanics, is demonstrably recovered when considering the distinguished leaf where the Hamiltonian commutes with itself, providing a crucial link to established theory. However, the innovation lies in extending this framework to encompass generic states, organised according to their corresponding leaf label, thereby establishing a structure suitable for investigating thermalisation beyond equilibrium.
This approach facilitates the formulation of a “leaf typicality” hypothesis, positing that local observables depend solely on the energy and the specific leaf the system occupies. To test this hypothesis, the research focuses on evolving representative pure states drawn from the optimal ensemble constructed on each leaf. This methodology deliberately avoids reliance on ensemble averaging, instead examining the behaviour of individual states to determine if they accurately reproduce the predictions of the leaf-canonical ensemble at all times.
The choice of pure states, rather than mixed states, is motivated by the desire to isolate and track the evolution of coherence within the system, providing a more sensitive probe of thermalisation dynamics. Scientists have long sought to reconcile the seemingly contradictory worlds of quantum coherence and thermal equilibrium. This new work offers a compelling framework for understanding how systems can appear to thermalize, reaching a stable, predictable state, without fully sacrificing the delicate quantum correlations that define their behaviour.
The challenge lies in the fact that standard statistical mechanics assumes a loss of coherence, yet coherence is crucial for many quantum technologies and fundamental physical processes. By introducing the concept of “minimum-variance leaves” within the system’s state space, researchers have devised a way to organize quantum states based on their energy fluctuations and coherence properties.
This suggests a deeper principle at play, termed “leaf typicality”, where local observable properties depend only on the energy and the specific leaf the system occupies. This offers a potential pathway beyond the limitations of traditional eigenstate thermalization theory, which struggles to fully account for systems far from equilibrium. However, the degree to which this framework holds true appears sensitive to the level of energy incoherence within the system, with convergence to predictable behaviour slowing as incoherence increases.
The demonstrated breakdown of leaf typicality in integrable systems, those with an abundance of conserved quantities, highlights a key limitation. Future work will likely focus on extending this approach to more complex, many-body systems and exploring the interplay between coherence, thermalization, and the emergence of macroscopic behaviour.