Informational Geodesic Collapse: Dynamic Reduction of the Quantum State Governed by the Curvature of the Fisher–Bures Metric in Entangled Subsystems

Abstract

We propose the Informational Geodesic Collapse Theory (IGCT), a stochastic extension of the projection postulate of quantum mechanics in which the rate of dynamical reduction of the state vector is not a universal parameter coupled solely to the system’s mass — as in Continuous Spontaneous Localization (CSL) and Ghirardi–Rimini–Weber (GRW) models — but a geometrically defined quantity, computed from the Ricci scalar curvature of the Bures–Uhlmann metric induced on the manifold of reduced states of a subsystem, weighted by its von Neumann entanglement entropy. The construction builds upon the already established formalism of the geometrization of quantum mechanics (projective Hilbert space as a Kähler manifold, unitary dynamics as a Hamiltonian flow) and grafts onto it a stochastic diffusion term whose coefficient — the informational curvature scalar KA(ψ) — vanishes identically for product states and grows with the degree of entanglement of the considered subsystem. We rigorously demonstrate (i) that the proposed stochastic equation preserves the norm of the state vector for any nonnegative and adapted choice of KA(ψt); (ii) that, in the limit where KA is held constant, the theory reduces exactly to the standard CSL model, inheriting its linear master equation and its proof of no superluminal signaling; (iii) that, for a two-level subsystem, the Bures metric is a space of constant scalar curvature (R = 24, isometric to a hemisphere of S 3 of radius 1/2), a result obtained by direct symbolic computation, from which it follows that KA is, in this case, strictly proportional to the entanglement entropy; and (iv), as a novel extension of this work, that for a three-level subsystem (qutrit) the scalar curvature of the isospectral leaf of the Bures metric — the flag manifold SU(3)/U(1)2 equipped with the induced metric — is not constant, having been exactly computed (via Lie algebra verified symbolically and numerically validated against the qubit case) at representative points of the eigenvalue space, with exact values between ≈ 191 and ≈ 3120 (in natural units) and a divergence as the state approaches the maximally mixed state. Complementarily, we perform a Monte Carlo stochastic simulation of Eq. (6) that numerically confirms, with excellent agreement with the analytical prediction e −2λKAt, that a highly entangled subsystem (Bell pair, KA = 24 ln 2) loses interference visibility at a rate about 330 times larger than a subsystem of identical mass and residual entanglement, numerically isolating the theory’s central phenomenological signature. Thus, the theory predicts an objective collapse rate that depends not only on mass but on the internal entanglement structure of the system — an experimentally distinguishable signature from GRW/CSL/Diósi–Penrose models, testable in optomechanical interferometry with controlled pre-entanglement. We honestly discuss the limitations of the proposal: the absence of superluminal signaling in the fully state-dependent regime remains conjectural, a relativistically covariant generalization is not constructed here, and the exact computation Informational Geodesic Collapse 2of R[gBures] for d ≥ 3 remains restricted, in this version, to the isospectral leaf (the full flag manifold of D(HA) in dimension 3, including the radial directions of the spectrum, has not been computed). The IGCT is resented as a mathematically well-posed and falsifiable research program, not as an established fact of nature.