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.
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 develop a unified geometric framework for local non-adiabaticity in driven quantum systems. We show that the previously introduced AMT non adiabaticity parameter arises as a special realization of a more general geometric instability criterion governed by the normalized Fubini Study distinguishability speed. The local geometric evolution speed is identified as the physically relevant quantity controlling the onset of non-adiabatic instability.
We introduce a universal dimensionless instability parameter measuring the competition between quantum-state evolution speed and spectral-gap protection. This quantity provides a local, gauge-invariant, and basis-independent criterion for arbitrary driven Hamiltonians. Near quantum critical points, the instability parameter diverges through inverse gap amplification, recovering the Kibble Zurek freeze-out condition directly from local geometric data.
We prove that monotonic occupation-dependent nonlinear regulators geometrically compress the quantum metric, establishing a self-limitation theorem in which nonlinear spectral deformation confines the accessible region of projective Hilbert space under strong driving. The multimode extension yields a matrix-valued instability criterion that identifies collective instability channels invisible to scalar descriptions.
The framework naturally extends to open quantum systems through the Bures metric and quantum Fisher geometry, where thermal mixing and Lindblad decay increase the instability threshold through geometric suppression of state distinguishability. The instability threshold further implies a universal geometric lower bound on coherent control time and quantum gate duration.
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 develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary sufficiently slow protocols using adiabatic perturbation theory, and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those which drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses, or target individual excited states in quantum chaotic systems, the geometric bound for the QSL can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in non-integrable spin chains using optimal control.
The quantum adiabatic theorem conventionally posits that a quantum system initialized in an instantaneous eigenstate of a time-dependent Hamiltonian will remain in that corresponding instantaneous eigenstate, provided the parameter changes are sufficiently slow and no level crossings occur. Under this idealized framework, standard quantum mechanics neglects any intrinsic energetic cost associated merely with traversing a trajectory through the projective Hilbert space or the underlying parameter manifold, assuming that adiabatic evolution is inherently frictionless and dissipation-free in closed systems. In this work, we challenge this traditional paradigm by introducing the concept of Quantum-State Path Friction (QSPF). We propose that even under strictly ideal adiabatic conditions, an irreducible energetic penalty exists, which scales quadratically with the velocity of the quantum state along its geometric path. We formally define the path work as W_path = \int \eta_Q (ds_Q / dt)^2 dt, where ds_Q^2 = g_{ij} d\lambda_i d\lambda_j denotes the quantum geometric line element governed by the quantum metric tensor g_{ij}, and \eta_Q represents the quantum friction coefficient. This formulation establishes a foundational framework for a new class of dissipation laws in quantum dynamics, bridging quantum information geometry, non-equilibrium thermodynamics, and optimal quantum control.
Abstract We propose physical distinguishability as the fundamental primitive underlying quantum mechanics, general relativity, and thermodynamics. A state is defined operationally: two states are physically distinct if a physical interaction or measurement can reveal a difference between them. From this single principle, the distinguishability divergence D generates a smooth state space endowed with a Riemannian metric , a geometric velocity, an accessibility scale Λ, an accessibility ratio , and a geometric time . The framework provides a unified geometric description of the three major domains of physics. Quantum mechanics emerges through the Fubini–Study geometry of projective Hilbert space, thermodynamics through Fisher–Rao and Ruppeiner information geometry, and general relativity through the geometry of geodesic deviation. The threshold condition A=1acquires a universal physical meaning and appears in all three realizations as the boundary between physically accessible and inaccessible evolution. Variational formulations of both the Einstein and Schrödinger equations are constructed within the same geometric framework. The Lorentzian signature of spacetime emerges from a single sign reversal associated with the geometric-time direction, while the coupling constant of the distinguishability action is connected to the observed vacuum energy density. The framework further predicts a characteristic vacuum distinguishability scale ≈8.8×10^(-5) m, corresponding to an energy scale≈2.25" " meV. The theory is tested experimentally on IBM Quantum superconducting processors. Across one-qubit, two-qubit, 127-qubit, and 156-qubit experiments performed on three independent Heron-generation backends, excitation probabilities collapse onto universal curves when parameterized by endpoint Fubini–Study distinguishability. Protocols with identical laboratory control parameters but different endpoint distinguishabilities produce excitation probabilities differing by up to a factor of 3.8, while protocols with identical distinguishability produce identical outcomes across independent processors. Cross-backend transfer errors remain below 1%, providing direct experimental support for endpoint distinguishability as a universal physical variable. Taken together, the theoretical construction, variational formulations, vacuum-sector predictions, and large-scale quantum-processor experiments support distinguishability geometry as a unified and experimentally testable foundation for physical dynamics.