We show that isotropic random walks on the complex projective space provide a canonical and analytically tractable stochastic-geometric framework for the exploration of quantum-state space. The approach combines harmonic analysis on compact rank-one symmetric spaces with stochastic pure-state evolution and yields explicit analytical expressions for transition kernels, fidelity statistics, and geometric observables associated with the Fubini--Study metric. In particular, the framework provides a solvable reference model for isotropic depolarization and Haar equilibration, reproducing Haar-random fidelity statistics and the invariant measure on projective Hilbert space without specifying a microscopic Lindblad generator. In the short-time regime, the stochastic evolution converges to Brownian diffusion generated by the Fubini--Study Laplace--Beltrami operator, while the long-time limit exhibits concentration-of-measure behaviour characteristic of high-dimensional random quantum states. We further derive analytical and asymptotic results for the first-passage-time problem, including closed-form expressions in the Brownian limit for the mean first passage time and the long-time tail of the first-passage-time distribution. For high-fidelity target states, the mean first passage time exhibits a strong dimension-dependent divergence originating from the concentration properties of the Fubini--Study geometry.
We show that isotropic random walks on the complex projective space provide a canonical and analytically tractable stochastic-geometric framework for the exploration of quantum-state space. The approach combines harmonic analysis on compact rank-one symmetric spaces with stochastic pure-state evolution and yields explicit analytical expressions for transition kernels, fidelity statistics, and geometric observables associated with the Fubini--Study metric. In particular, the framework provides a solvable reference model for isotropic depolarization and Haar equilibration, reproducing Haar-random fidelity statistics and the invariant measure on projective Hilbert space without specifying a microscopic Lindblad generator. In the short-time regime, the stochastic evolution converges to Brownian diffusion generated by the Fubini--Study Laplace--Beltrami operator, while the long-time limit exhibits concentration-of-measure behaviour characteristic of high-dimensional random quantum states. We further derive analytical and asymptotic results for the first-passage-time problem, including closed-form expressions in the Brownian limit for the mean first passage time and the long-time tail of the first-passage-time distribution. For high-fidelity target states, the mean first passage time exhibits a strong dimension-dependent divergence originating from the concentration properties of the Fubini--Study geometry.
We develop a geometric formulation of classical and quantum dynamics in which projective Hilbert space is taken as the primary state space and classical configuration and phase spaces arise as submanifolds of localized quantum states. The Fubini-Study metric induces the Euclidean metric on these submanifolds, while the tangential component of Schrödinger evolution reproduces Hamiltonian and Newtonian dynamics. We then extend the construction to stochastic motion. Brownian dynamics on the localized classical manifold is lifted to a common stochastic unitary evolution in projective Hilbert space; under homogeneity, isotropy, and independence assumptions, the corresponding Hermitian increments have Gaussian Unitary Ensemble statistics. This yields a random-matrix Schrödinger dynamics whose transition probabilities depend only on Fubini-Study distance. Through the embedding of localized states, the classical Gaussian transition law becomes the Born rule in projective state space. Measurement outcomes are identified with first arrival at detector-defined finite-resolution equivalence classes, while each realized state-space evolution remains linear and unitary. The framework is applied to classical emergence, macroscopic stabilization, double-slit experiments, composite systems, and entangled states. A spatially local Caldeira-Leggett-type system-environment model is also shown, under explicit thermal, symmetry, Ohmic, and coarse-graining assumptions, to generate the Brownian family required by the stochastic lift. The remaining open problem is to establish the range and universality of this mechanism for realistic many-body measuring environments.
We develop a state-space geometric framework for measurement, classicality, and quantum paradoxes, based on one dynamical conjecture. Classical configuration space and classical phase space for a mechanical system arise as distinguished submanifolds of projective quantum state space. On these submanifolds, the Fubini--Study geometry induces Euclidean classical geometry, and the tangent component of Schrödinger evolution reproduces Newtonian dynamics. Within this framework, interactions with measuring devices and environments are described by random-matrix dynamics on projective state space, generated by matrices drawn from the Gaussian Unitary Ensemble. We show that this random-matrix dynamics yields isotropic diffusion, giving Born-rule transition probabilities in microscopic measurements and stabilizing classical behavior in macroscopic systems. We further argue that the random-matrix conjecture is not an independent ad hoc assumption: under natural translation-invariance assumptions on the distribution of state-space steps originating on the classical submanifold, the unitary lift of homogeneous and isotropic Brownian motion on that submanifold is uniquely given by the Gaussian Unitary Ensemble, up to scale and an irrelevant scalar part. The resulting framework provides a unitary account of measurement and the quantum-to-classical transition and, if accepted, offers a dynamical resolution of standard quantum paradoxes.
Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.
First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
We present numerical simulations supporting the random-matrix state-reduction framework developed in the companion theoretical paper. The simulations test the main derived features of the model: isotropic diffusion generated by Gaussian Unitary Ensemble Hamiltonians in projective state space, the restriction of this diffusion to Brownian motion on the classical submanifold, Born-rule frequencies for detector-defined outcome classes, and stroboscopic Newtonian motion for macroscopic systems under repeated environmental monitoring. We also compare GUE and GOE random Hamiltonians and show that GOE fails to produce the required isotropic complex projective diffusion. Further simulations examine finite-resolution detector records in the double-slit experiment, Zeno stability of recorded equivalence classes, effective irreversibility from high-dimensional state-space dynamics and loss of path information, and tensor-product particle-device dynamics in the device limit. The results show that microscopic state reduction, stable measurement records, effective irreversibility, and macroscopic classicality can be described as different coarse-grained manifestations of the same stochastic unitary mechanism.
Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert–space formulation of classical statistical thermodynamics, we show that the specification of a canonical polarity on the real projective space RPn induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, equilibrium thermal states are determined by a Hamiltonian gradient flow with respect to this metric. This flow is characterized by the property that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is then extended to establish a new framework for the quantum–mechanical dynamics of equilibrium thermal states. In this case, the relevant phase space is the complex projective space CPn, here regarded as a real manifold γ endowed with the Fubini–Study metric and a compatible symplectic structure. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the non–uniqueness of the infinite–temperature state. We are then led to formulate a geometric characterization of the standard KMS relation often considered in the context of C* algebras. Finally, we develop a theory of the quantum microcanonical and canonical ensembles, based on the geometry of the quantum phase space γ. The example of a particle of quantum spin–1/2 particle in a heat bath is studied in detail.