We formulate a geometric extension of open quantum dynamics in which the density matrix evolves as a Riemannian gradient flow on the Bures manifold. The evolution equation \frac{d\rho}{dt} = -i[H,\rho] - \lambda \nabla_B \Phi[\rho] drives the state toward regions of minimal information degradation, defined by the functional \Phi[\rho], the instantaneous rate of relative-entropy change under the bare Lindbladian. This deterministic flow preserves trace and positivity, and introduces a single new universal constant \lambda>0 while recovering standard Lindblad dynamics as \lambda\to0. Under a Unitary Orthogonality Condition – exactly satisfied for pure dephasing and approximately in many-body localized phases – \Phi becomes a Lyapunov functional, and its critical points (decoherence-free subspaces) are asymptotically stable attractors. Applied to disordered spin chains, the framework reproduces the known entropy‑suppression scaling \Phi \lesssim \gamma N (J/W)^2. The theory offers a falsifiable, information‑geometric principle where quantum states intrinsically minimise their own information loss. Keywords: quantum information geometry, Bures metric, gradient flow, Lindblad dynamics, decoherence-free subspaces, many-body localization, Lyapunov functional
We formulate a geometric extension of open quantum dynamics in which the density matrix evolves as a Riemannian gradient flow on the Bures manifold. The evolution equation \frac{d\rho}{dt} = -i[H,\rho] - \lambda \nabla_B \Phi[\rho] drives the state toward regions of minimal information degradation, defined by the functional \Phi[\rho], the instantaneous rate of relative-entropy change under the bare Lindbladian. This deterministic flow preserves trace and positivity, and introduces a single new universal constant \lambda>0 while recovering standard Lindblad dynamics as \lambda\to0. Under a Unitary Orthogonality Condition – exactly satisfied for pure dephasing and approximately in many-body localized phases – \Phi becomes a Lyapunov functional, and its critical points (decoherence-free subspaces) are asymptotically stable attractors. Applied to disordered spin chains, the framework reproduces the known entropy‑suppression scaling \Phi \lesssim \gamma N (J/W)^2. The theory offers a falsifiable, information‑geometric principle where quantum states intrinsically minimise their own information loss. Keywords: quantum information geometry, Bures metric, gradient flow, Lindblad dynamics, decoherence-free subspaces, many-body localization, Lyapunov functional
We develop a geometric theory of states arising from the Universal Modular Dynamics (UMD) framework, in which the density operator ρ serves as the fundamental object and geometry is induced by the modular generator K = −log ρ. Within this construction, a Riemannian structure on state space is defined via information-geometric relations, leading to a natural notion of curvature and variational dynamics. We introduce a variational action functional S[ρ] built from geometric and informationtheoretic contributions, and show that it induces a gradient flow on the space of states. This flow defines an intrinsic, irreversible dynamics characterized by a monotonic decrease of S, establishing it as a Lyapunov functional and providing a geometric origin for the arrow of time. Numerical analysis demonstrates that the resulting dynamics generically suppresses offdiagonal coherence and drives states toward a universal attractor corresponding to the maximally mixed configuration. This attractor is shown to be stable under perturbations and independent of initial conditions, thereby identifying classicality as a dynamically emergent and geometrically preferred phase. Importantly, the framework does not rely on external environments, measurement postulates, or stochastic noise. Instead, decoherence and equilibration arise intrinsically from the variational geometry of states defined by UMD. The results suggest that classical behavior can be understood as a universal outcome of geometric state dynamics, providing a novel perspective on irreversibility and the emergence of classicality in quantum systems.
This work investigates the emergence of state-dependent dissipative structures in open quantum systems and develops a geometric framework for constructing physically admissible non-linear extensions of Markovian quantum dynamics. Building upon the coarse-grained completely positive generators developed in previous work, we consider the effects of finite and dynamically polarizable environments, leading naturally to state-dependent Kossakowski structures and non-linear dissipative flows. To constrain such non-linear dynamics and prevent the appearance of unphysical states, we formulate the evolution as a gradient flow on the manifold of quantum states endowed with the Bures metric. A locality-preserving Extension Consistency Condition is introduced to ensure the admissibility of non-linear lifts to composite systems and to eliminate non-local ancillary back-action. Within this framework, we establish a Boundary Repulsion Theorem showing that the rank-deficient boundary of the positive state manifold is located at an infinite geometric distance along the flow trajectories. As a consequence, positivity of the density operator is preserved for all finite evolution times. A worked example based on a three-level V-system demonstrates the regularizing effect of the Bures geometry in the presence of state-dependent dissipation. The results suggest that information-geometric structures may provide a natural route toward physically consistent non-linear quantum master equations while preserving the operational integrity of quantum state evolution.
We develop a geometric theory of states arising from the Universal Modular Dynamics (UMD) framework, in which the density operator ρ serves as the fundamental object and geometry is induced by the modular generator K = −log ρ. Within this construction, a Riemannian structure on state space is defined via information-geometric relations, leading to a natural notion of curvature and variational dynamics. We introduce a variational action functional S[ρ] built from geometric and informationtheoretic contributions, and show that it induces a gradient flow on the space of states. This flow defines an intrinsic, irreversible dynamics characterized by a monotonic decrease of S, establishing it as a Lyapunov functional and providing a geometric origin for the arrow of time. Numerical analysis demonstrates that the resulting dynamics generically suppresses offdiagonal coherence and drives states toward a universal attractor corresponding to the maximally mixed configuration. This attractor is shown to be stable under perturbations and independent of initial conditions, thereby identifying classicality as a dynamically emergent and geometrically preferred phase. Importantly, the framework does not rely on external environments, measurement postulates, or stochastic noise. Instead, decoherence and equilibration arise intrinsically from the variational geometry of states defined by UMD. The results suggest that classical behavior can be understood as a universal outcome of geometric state dynamics, providing a novel perspective on irreversibility and the emergence of classicality in quantum systems.
The onset of decoherence in open many-body systems lacks a dynamical timescale grounded in the loss of bipartite entanglement. Here, we introduce the geometricdecoherencetime, defined as the earliest moment the monotone relation between logarithmic negativity and Rényi-21 entropy -- exactly equal across any bipartition for pure states -- breaks down under open-system evolution, signaling entropy growth without accompanying entanglement growth. We establish this criterion in both single-particle Gaussian dynamics and many-body Lindbladian evolution. We show that quantum mutual information provides a complementary long-time diagnostic: its asymptotic vanishing is equivalent to factorization of the steady state across the bipartition, a condition strictly stronger than separability, and whenever a product steady state is approached exponentially in trace norm, negativity and mutual information share the same decay rate. In the presence of a strong symmetry, this tracking can fail -- residual classical correlations can survive after entanglement has vanished. In the Kitaev chain with balanced gain and loss, we derive a closed-form solution and show that the topological phase sustains longer coherence times than the trivial phase at identical dissipation, with a local minimum at the chiral-symmetric point. In the interacting XXZ chain, exact many-body evolution shows that local Z-dephasing preserves residual classical correlations, whereas gain and loss restore the mutual-information tracking of negativity. Our results establish the geometric decoherence time as a dynamical scale tracking the onset of decoherence.
We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
The second law does more than set the direction of thermodynamic evolution: it endows nonequilibrium transformations with an underlying geometry. In this work, we provide a unified geometric description of entropy production in classical and quantum thermodynamics based on Wasserstein-2 structures arising from gradient flows of free energy. We review how relaxation to equilibrium, in overdamped diffusions, discrete detailed-balanced Markov chains, and dissipative Lindblad dynamics, can be formulated as a gradient flow on the space of states. The associated Wasserstein-2 distance bounds entropy production, yielding a finite-time refinement of the second law. We extend this framework beyond purely dissipative dynamics by introducing generalized Wasserstein-2 metrics that incorporate conservative (Hamiltonian) dynamics in both classical inertial systems and open quantum systems, yielding intrinsic distances that exactly characterize minimal entropy production under fixed dissipative mobilities. We establish equivalence bounds between purely dissipative and Hamiltonian-dissipative geometries, explicitly quantifying how inertial or coherent dynamics can reduce dissipation. Finally, when restricted to equilibrium distributions, we recover the thermodynamic length of linear response-including the quantum thermodynamic length-thereby linking optimal transport, thermodynamic length, and counterdiabatic protocols within a single geometric framework. All in all, our results extend the Riemannian program of thermodynamics further from equilibrium and provide a geometric foundation for optimal protocols beyond the overdamped setting.