This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.
This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.
We extend the geometric tracking infrastructures established in preceding works to formalize the strict coordinate and dynamical limitations that govern parameterized quantum steering protocols. By constructing a differential-geometric framework mapping smooth control parameters directly to the projective Hilbert space, we isolate the architectural conditions under which optimization protocols undergo structural failure. First, we state and prove the Control Immersion Criterion, demonstrating that any trajectory intersecting a localized coordinate caustic interface triggers a sharp rank deficiency in the pulled-back information metric tensor. This metric non-invertibility forces a catastrophic, asymptotic divergence in the local Levi Civita Christoffel connection symbols (Γmij → ∞), completely destabilizing gradient-driven updates. Second, we evaluate the global phase-space volume element under cyclic parameters, proving that the anti-Hermiticity of the connection generator ensures a strictly real-valued Berry connection and subsequent structural closure of the modified symplectic form (dω=0), maintaining topological conservation despite severe localized metric warping. Finally, we derive a sharp kinematic speed limit bounding non-adiabatic escape rates out of targeted operating subspaces as a direct penalty scaled by the metric tensor. Crucially, this bound exposes an operational duality linking statistical estimation constraints to state-leakage rates through the Quantum Fisher Information Metric structure (gQF IM = 4gij ). Dense grid calculations and dynamic fourth-order Runge-Kutta numerical simulations confirm these analytical constraints down to the machine precision floor, validating this framework for structural pre-mapping routines inside advanced, singularity-aware quantum compilation engines.
This paper develops a rigorous, self-contained differential-geometric framework for tracking mixed quantum states in open physical systems, providing an analytical structural extension to the pure-state Kâhler geometry of Paper 1. Because open quantum dynamics induced by environment-driven dissipation and decoherence break the global manifold homogeneity, we model the total space of density operators S(H) as a stratified space. We prove that the individual rank-k subsets constitute embedded smooth real manifolds of dimension 2k(N + 1) − k2 − 1, satisfying the algebraic conditions of Whitney Regularity Condition B to guarantee continuous tangent space alignment across interfaces. To assign coordinate-independent connection and metric tensors over this stratified landscape, we formalize the global principal U (n) fiber bundle structure using Uhlmann’s purification theorem. By lifting mixed density operators to pure-state amplitudes W ∈ GL(n, C) on a bipartite workspace, we show that minimizing the ambient flat Hilbert–Schmidt metric along the vertical fibers induces the non-degenerate Bures metric on the base manifold. We derive the exact matrix elements of the corresponding Symmetric Logarithmic Derivative (SLD) fields in an arbitrary non-degenerate eigenbasis, establishing the mixed-state information scaling boundary gQF IM = 4gB. Finally, we characterize the non-Abelian Yang–Mills curvature FU = dAU + AU ∧ AU of the Uhlmann connection, demonstrating how non-commutativity emerges naturally under environmental coupling. By isolating the measure-zero boundary configurations where eigenvalues or matrix rank drop, we generalize our control immersion criterion to identify both rank deficient and eigenvalue-vanishing caustic singularities. This metric-determinant verification pipeline allows researchers to systematically diagnose, map, and structurally bypass trajectory bottlenecks during Lindbladian control path optimization on real-world noisy quantum backends
We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point ω=0 and g/Γ=1/2; the physical manifold is non-Kähler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring Kähler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.
Abstract One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher–Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states, the Fubini–Study tensor, which is preserved under unitary evolution. In this work, we explore how the modification of the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini–Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.
Abstract Driven isolated quantum systems lack a universal local energy scale that characterizes the instantaneous intensity of coherent nonequilibrium evolution. Existing descriptions rely on transition probabilities, spectral gaps, excitation densities or long-time effective ensembles, but none of these quantities is determined solely by the geometry of the evolving quantum trajectory. Here we show that requiring locality, gauge invariance, basis independence and the correct energy dimension is sufficient to fix such a scale uniquely among first-order local scalars built from the unitarily invariant Riemannian line element of projective Hilbert space, and that projective Hilbert-space geometry provides it: for any smooth pure-state unitary evolution, the instantaneous Fubini–Study speed v_FS generates a local geometric energy E_geo=ℏv_FS, which is gauge invariant, basis independent and determined only by the local trajectory of the state. The same scale is the local fidelity-loss rate, coincides with the quantum-speed-limit energy scale, and governs the onset of non-adiabatic excitation. Beyond this energetic characterization, the hyperbolic structure of the evolution generator gives this energy a distinct thermodynamic representation, k_B T_geo=ℏκ_g/2π, whose 2π normalization is derived — not postulated — from the exact thermal scattering structure of the generator, by the same analytic-continuation mechanism that fixes the Unruh and Hawking temperatures. The resulting temperature saturates the Maldacena–Shenker–Stanford bound on chaos algebraically, a formal correspondence of the linearized single-mode dynamics rather than a demonstration of many-body chaos. Our results establish intrinsic geometric energy as a state-evolution variable of driven quantum systems and provide a common foundation for non-adiabatic excitation, geometric self-limitation and dynamical bounds on information scrambling.
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space.
We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes.
Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure.
These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.