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 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.
This paper extends Uhlmann's parallel-transport geometry for mixed quantum states beyond the conventional full-rank setting to the Global Hybrid Operator Density of a GENERIC-consistent Gay–Balmaz–Tronci hybrid quantum–classical closure, building on the well-posedness and thermodynamic-consistency results established in two companion papers. The central object is a Hybrid Uhlmann connection whose holonomy around a closed recovery cycle yields three main results: the Global Hybrid Winding Theorem, showing that the holonomy spectrum changes if and only if a classical-sector memory kernel is genuinely non-Markovian (sharpened to an exact, non-perturbative statement for a fixed physical direction); an Abelian Blindness theorem, proving the holonomy's determinant channel is exactly and unconditionally insensitive to occupation loss (leakage), even as the underlying state approaches or crosses the rank-deficient boundary; and a Directional Singularity theorem, showing that the full non-Abelian spectrum instead develops a genuine, non-removable singularity there, with the limiting value depending on the direction of approach to the boundary. Together these results show that a single geometric quantity — the holonomy spectrum — can separate non-Markovian memory from leakage, two error mechanisms that existing diagnostics require distinct, purpose-built tools to detect. The theorems are verified on a structure-preserving numerical integration scheme and accompanied by a concrete, falsifiable experimental protocol for superconducting qutrit hardware, extending a recently demonstrated scalar Uhlmann-phase measurement to full non-Abelian holonomy reconstruction.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
We work in the universal information hydrodynamics (UIH) framework developed in two companion papers, where reversible quantum dynamics and irreversible response are organised in a local metriplectic geometry generated by a convex free energy, a Fisher metric, and an antisymmetric no work channel. In this setting the irreversible drift is always a Fisher gradient flow in a weighted
H−1 geometry, equipped with a cost entropy inequality and a curvature coercivity bound that are invariant under the addition of any compatible reversible dynamics. In the present paper we show that this structure has a canonical realisation for Fokker Planck equations, finite Markov chains and GKLS semigroups, and that it can be reconstructed operationally from process and state tomography on real quantum hardware. First, starting from overdamped Langevin dynamics and its Fokker Planck generator, we identify a canonical Fisher Dirichlet operator
G=Qdiag(π) that plays the role of the symmetric mobility on a periodic lattice and saturates the cost entropy inequality mode by mode. We then demonstrate that in a wide class of thermal and driven GKLS models the symmetric part of the real generator in the stationary BKM metric has a density block that coincides exactly with this classical Fisher operator, including nonreversible chains and random qubit and qutrit ensembles, and that the global decay of entropy and Fisher energy is governed by the Markov spectral gap while the Fisher gap controls only local curvature. Finally, we implement a suite of IBM Quantum experiments on a superconducting qubit that realises the full UIH picture on hardware. Process tomography of idle circuits yields an effective real generator
K in the Pauli basis whose metric adjoint split in the BKM metric at the device stationary state produces a clean metriplectic decomposition
K=G+J, with
G metric symmetric and
J metric skew. Independent idle depths are consistent with a single time homogeneous generator, the dissipative spectrum of
G sets an information theoretic speed limit for BKM Fisher decay, and a curvature test shows that the same BKM metric is the local Hessian of quantum relative entropy at the stationary state. All claims are backed by a reproducible Python archive that includes constructive Fokker Planck to Markov to GKLS lifts, hypocoercive decay diagnostics, and hardware K tomography. Taken together, the results in this paper show that this programme is not only mathematically coherent, but experimentally visible, showing strong signatures of universality.