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Well-Posedness of the Gay-Balmaz–Tronci Hybrid Quantum–Classical Closure

Louis Nguyen

Published Jul 25, 2026• 0 citations
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Abstract

This paper establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure introduced by Gay-Balmaz and Tronci. By constructing an admissible operator-valued state space together with a vacuum control condition that resolves the structural singularity at vanishing classical density, the work proves local existence, uniqueness, and a Beale–Kato–Majda-type continuation criterion, together with sufficient conditions for global-in-time existence under physically relevant hybrid Hamiltonians. Numerical verification across fifteen representative quantum-computing platforms confirms preservation of the hybrid closure, operator positivity, and bounded continuation behavior. These analytical results provide the mathematical foundation for subsequent thermodynamic, geometric, and topological extensions of hybrid quantum–classical dynamics, including the irreversible GENERIC framework developed in the companion paper

DOI
10.5281/zenodo.21577563

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This paper establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure introduced by Gay-Balmaz and Tronci. By constructing an admissible operator-valued state space together with a vacuum control condition that resolves the structural singularity at vanishing classical density, the work proves local existence, uniqueness, and a Beale–Kato–Majda-type continuation criterion, together with sufficient conditions for global-in-time existence under physically relevant hybrid Hamiltonians. Numerical verification across fifteen representative quantum-computing platforms confirms preservation of the hybrid closure, operator positivity, and bounded continuation behavior. These analytical results provide the mathematical foundation for subsequent thermodynamic, geometric, and topological extensions of hybrid quantum–classical dynamics, including the irreversible GENERIC framework developed in the companion paper
This manuscript establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure of Gay-Balmaz and Tronci. The central result, the Covariance-Symplectic Obstruction Theorem, shows the uniformly hyperbolic admissible sector is generically empty, with local well-posedness instead established on a weaker "uniformly symmetrizable" sector via an explicit microlocal symmetrizer and a Friedrichs–Kato existence argument. A companion Rank-Deficiency Theorem classifies six standard device Hamiltonians (transmon-resonator, tunable coupler, Rydberg array, trapped-ion, optomechanical, and Kerr-cat architectures) as excluded from the hyperbolic sector, connecting the abstract theory to concrete qubit platforms.
This paper develops the first thermodynamically consistent GENERIC formulation of the Gay-Balmaz–Tronci hybrid quantum–classical closure by constructing a global hybrid operator density that unifies reversible geometric dynamics with irreversible GKSL and Fokker–Planck dissipation. It establishes rigorous disintegration and completeness theorems for the admissible state space, extends the Berry-curvature-corrected hybrid entropy to mixed conditional quantum states, and proves its Abelian gauge invariance on the non-degenerate admissible manifold. Under an explicitly stated well-posedness hypothesis, the fused hybrid dynamics is shown to satisfy all four GENERIC consistency conditions, while recovering both classical GENERIC and the standard GKSL/Spohn theorem as limiting cases. These results provide a rigorous mathematical framework for nonequilibrium hybrid quantum–classical systems and lay the thermodynamic foundation for future studies of correlated and non-Markovian noise in open quantum devices.
We develop a conditionally closed reconstruction of quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime--vacuum substrate, and persistent causal source--response coupling. Wave functions, Hilbert spaces, complex amplitudes, density operators, the Born rule, commutation relations, Fock spaces, path integrals, completely positive maps, and Markovian dynamics are introduced only as derived mathematical or operational structures, not as additional physical axioms. The construction distinguishes the actual finite carrier and its causal history from the quantum state used to represent preparations, transformations, and measurement probabilities. It derives unitary dynamics from self-adjoint Hamiltonians, formulates spectral observables and probability currents, and states the explicit equilibrium, additivity, noncontextuality, and calibration conditions required for the Born representation. Measurement is described through instruments, pointer dynamics, stable records, weak values, continuous monitoring, diffusive and counting trajectories, and operational error--disturbance relations. Quantum trajectories remain record-conditioned predictive representations rather than automatically ontic paths. Composite systems, identical particles, spin, entanglement entropy, purification, steering, Bell correlations, dense coding, and no-broadcasting are developed on declared tensor-product and symmetry domains. Open-system dynamics are formulated through causal Volterra equations, memory kernels, initial-slip terms, auxiliary-mode realizations, and unitary dilations. Existence, positivity, complete positivity, trace preservation, no-signalling, and energy closure are treated as logically distinct properties. Semiclassical pseudodifferential calculus, controlled path-integral convergence, Ehrenfest-time bounds, semiclassical measures, decoherence, and macroscopic-record stability define the quantum--classical interface. Topological phases, defect-bound states, relativistic wave equations, infrared dressing, and effective quantum-field-theory domains are incorporated with explicit applicability and error conditions. Numerical spectral certification, statistical model comparison, twenty-five benchmark models, theorem-status ledgers, and a complete experimental statistical protocol provide falsification and replication criteria. The resulting framework preserves standard quantum mechanics on its validated domain while isolating finite-core and historical corrections as conditional, quantitatively bounded, and experimentally testable extensions. **Keywords** Quantum foundations; topological quantum mechanics; historical memory; three-axiom reconstruction; Born rule; quantum measurement; weak measurement; quantum trajectories; entanglement; spin and statistics; open quantum systems; Volterra equations; semiclassical limit; quantum topology; numerical certification; experimental falsification.
This paper develops a mixed-state geometric framework for hybrid quantum–classical dynamics by constructing a Hybrid Uhlmann connection on the Global Hybrid Operator Density of the GENERIC-consistent Gay-Balmaz–Tronci closure. Building on the Stiefel-bundle parallel transport formulation of Avitabile and Bridges together with an exactly unitary Cayley integrator, it proves the Global Hybrid Winding Theorem, establishing that the Uhlmann holonomy spectrum is invariant under memoryless evolution and changes if and only if the recovery cycle carries non-Markovian memory. The paper further characterizes the rank-deficient leakage boundary, proving that the determinant (abelian) channel is exactly blind to leakage while the full non-abelian holonomy develops an intrinsic directional singularity as occupations vanish. Numerical simulations illustrate these geometric signatures, and the paper proposes a general experimental readout protocol for future implementation across quantum computing architectures.
We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum–classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.
In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum–classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different “backaction” on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.