Thermodynamic completeness in quantum and classical Markovian dynamics

Abstract: We develop a path-space action formulation for quantum and classical Markovian thermodynamics that addresses a reconstruction problem: which thermodynamic observables can be inferred from state trajectories alone, and which require additional current or measurement record? The formulation treats the state trajectory and the thermodynamic record as distinct components of a Markovian path. In quantum systems, the record is specified by a quantum instrument; in the commutative classical representation, it is a density current constrained by a continuity equation. The main result is a thermodynamic completeness test. It identifies current or measurement-record perturbations that do not change the state trajectory and shows that any observable changed by such a perturbation cannot be reconstructed from state data alone. Hence two Markovian models can have the same state generator, stationary state, linear response, and local state geometry, but different thermodynamic current means or current noise. For quantum Markovian dynamics, an unconditioned density-matrix generator therefore does not determine heat-current, particle-transfer, photon-counting, spin-transfer, or continuous-measurement statistics; the quantum instrument and the thermodynamic increment assigned to each record outcome must also be specified. For classical density-current dynamics, the same test identifies hidden exchange, reaction, transport, and kinetic current records that are eliminated by the projection to the state trajectory. We further show that this incompleteness has geometric and topological origins: the current-space Hessian projects to a quotient state geometry, while graph cycles, divergence-free currents, and harmonic currents span directions invisible to state observations.

• The paper introduces a thermodynamic completeness test that distinguishes which observables are determined by state trajectories and which require explicit current information.
• It employs a unified path-space action framework using quantum instruments and classical continuity equations to differentiate observable statistics in Markovian models.
• The work reveals that systems with identical state generators can exhibit distinct current fluctuations, affecting experimental design and thermodynamic inference.

Thermodynamic Completeness in Quantum and Classical Markovian Dynamics

Path-Space Action Formalism and the Thermodynamic Reconstruction Problem

The paper establishes a unified path-space action framework for quantum and classical Markovian thermodynamics, focusing on a central operational question: which thermodynamic observables are reconstructible from state trajectories, as opposed to those requiring explicit knowledge of currents or measurement records? The formulation decouples the state evolution from the thermodynamic record, with the latter modeled by quantum instruments in quantum settings or density currents constrained by continuity equations in classical systems.

The central technical result is the thermodynamic completeness test. This criterion identifies which current or measurement record perturbations modify the record statistics without affecting the state trajectory. The main implication is that Markovian systems—even with identical state generators, stationary states, local response, and induced state geometry—can display distinct observable current means or noises due to unobservable directions in current space. Consequently, thermodynamic path observables such as heat or particle currents, full counting statistics, and entropy production are not determined solely by the state evolution.

Quantum and Classical Representations: Instruments, Currents, and Path Records

In the quantum case, the formulation distinguishes between the unconditioned evolution generated by a GKLS (Lindblad) operator and the thermodynamic record produced by a quantum instrument. Observables like photon-count, spin, or heat current statistics depend not only on the GKLS generator, but crucially on both the instrument specifying the record and the assignment of increments to each possible record outcome. This is formalized by identifying the structure of the quantum instrument space and the generator's decomposition into jump and diffusive measurement processes. The explicit construction demonstrates that two quantum Markov models with the same unconditioned generator can yield distinct statistics for any observable that does not factor through the state dynamics, such as heat-current or photon-counting cumulants.

In the classical setting, the analogous structure employs empirical or macroscopic currents associated with the Markov process, viewed as constrained by the continuity equation. The density-current action, defined via path-space relative entropy, centralizes the local information cost of a joint state-current trajectory. The operational implications mirror the quantum case: quantities dependent on the explicit current, such as exchange and cycle (loop) currents, require knowledge beyond state trajectories.

Geometric and Topological Non-Identifiability of Thermodynamic Records

The formalism reveals that non-identifiability of certain observables from state data alone is governed by both geometric and topological mechanisms. Geometrically, the local quadratic action for currents projects via a quotient map, defined by the kernel of the continuity operator (classical) or the quantum instrument map, to the observed state dynamics. State-induced geometries (e.g., Fisher, Kubo-Mori, Wasserstein, Onsager types) cannot reconstruct fluctuations in the hidden current directions.

Topologically, the space of state-invisible currents is identified with the cycle space of the transition graph (classical Markov networks) or, more generally, the divergence-free subspace (in continuous systems, via Hodge decomposition). Accordingly, observables associated with these cycles (e.g., cycle affinities, entropy production from circulation) are invisible to state-based measurements, but manifest in current statistics.

Observable Classification and Thermodynamic Action Classes

The analysis formalizes a precise classification of observables according to reconstructibility:

• State observables: Relaxation rates, susceptibilities, and local covariance, determined by the state action or quasipotential.
• Current observables: Mean and noise statistics of heat, particle, photon, reaction, or cycle currents, requiring the explicit path record unless the corresponding dual probe annihilates $\ker \Gamma_x$.
• Non-identifiable observables: Any for which the dual probe has nonzero projection on the kernel—distinct Markovian models with the same state dynamics but differing in current decomposition exist.

This separation is confirmed across three thermodynamic regimes—detailed-balance (reversible), driven non-equilibrium, and kinetic/Hamiltonian dynamics—each yielding different manifestations of hidden current structure. For example, in detailed-balance scenarios, local equilibrium statistics are state-determined, but exchange-current statistics are not. Driven systems can have identical state behavior with differing entropy production due to hidden cycle currents.

Implications and Future Directions

The path-space action-based reconstruction theory carries pronounced theoretical and experimental implications:

• Thermodynamic inference: No experimental protocol relying solely on state trajectory observations can infer observables sensitive to hidden current directions. Direct measurement or retention of the appropriate current/record is necessary for thermodynamic completeness.
• Model reduction and coarse-graining: Thermodynamic model reduction is well-posed only for observables surviving the state-current projection; otherwise, model ambiguity is intrinsic.
• Design of quantum and classical experiments: Choosing which observables are accessible determines whether full thermodynamic model specification is achievable, impacting calorimetric, transport, and quantum control experiments. In quantum scenarios, the necessity of specifying the quantum instrument for full thermodynamic description is made explicit.
• Geometric approach limitations: State-space geometry, no matter how refined, cannot capture noise or dissipation associated with unobservable currents, implying that fluctuation geometry is incomplete without explicit current information.
• Topological constraints on inference: Cycle-space dimensionality places upper bounds on the number of independently tunable current observables invisible to the state trajectory.

Possible future avenues include: generalizations to non-Markovian models with memory kernels, extensions to continuous control or parameter-dependent thermodynamics, investigation of metastable transitions and rare-event statistics beyond local fluctuation theory, and systematic usage of the completeness test in stochastic thermodynamic inference for partially accessible or coarse-grained systems.

Conclusion

The paper provides a rigorous, mathematically grounded operational criterion for thermodynamic completeness in quantum and classical Markovian dynamics, formalizing the conditions under which thermodynamic observables are determined by state trajectories or require explicit current or measurement records. The results delineate sharp boundaries for thermodynamic inference, model reduction, and the interpretation of experimental data, reinforcing the non-interchangeability of state and current observables in stochastic thermodynamics.

Notable claims include the provable existence of Markovian models with identical state-level dynamics but distinct current-level statistics, necessitating current- or record-level information for full thermodynamic specification. This represents a decisive step in clarifying the operational significance of path-space thermodynamics and the intrinsic limitations of state-based descriptions in the analysis of nonequilibrium stochastic systems.

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Reference:

"Thermodynamic completeness in quantum and classical Markovian dynamics" (2605.02650)