Quantum-geometric speed and dynamical instability are two natural rates for a driven quantum system, and their relation is unsettled even for exactly solvable dynamics. Here we show that for any multimode quadratic bosonic system referred to the bare-mode vacuum the Fubini-Study speed v_FS is the Frobenius norm of the symmetric, stretching part of the flow. This yields a sharp bound, lambda_max <= sqrt(2) v_FS, saturated by a resonant pure squeezer. The bound is not invertible: in a detuned parametric amplifier we hold either rate fixed while varying the other.
Quantum-geometric speed and dynamical instability are two natural rates for a driven quantum system, and their relation is unsettled even for exactly solvable dynamics. Here we show that for any multimode quadratic bosonic system referred to the bare-mode vacuum the Fubini-Study speed v_FS is the Frobenius norm of the symmetric, stretching part of the flow. This yields a sharp bound, lambda_max <= sqrt(2) v_FS, saturated by a resonant pure squeezer. The bound is not invertible: in a detuned parametric amplifier we hold either rate fixed while varying the other.
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.
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.
Bosonic Bogoliubov-de Gennes (BBdG) Hamiltonians describe the excitations of weakly interacting Bose condensates as well as photonic systems under parametric driving. Their topological features have been studied mainly by utilizing a generalized symplectic version of the Berry curvature and related Chern numbers. However, a full characterization of geometrical features in BBdG systems is still lacking. Here, we propose a symplectic quantum geometric tensor (SQGT), whose imaginary part leads to the previously studied symplectic Berry curvature, while the real part gives rise to a symplectic quantum metric, providing a natural distance measure in the space of bosonic Bogoliubov modes. The SQGT is directly related to observable properties of BBdG systems. We show how to measure all components of the SQGT by extracting excitation rates in response to periodic modulations of the system’s parameters. Moreover, we connect the symplectic Berry curvature to a generalized symplectic anomalous velocity term for Bogoliubov-Bloch wave packets. We test our results for a bosonic Bogoliubov-Haldane model.
We formalize the exact optimization boundaries, physical time evolution minimums, and finite-resource measurement constraints that govern the performance of the Quantum Natural Gradient (QNG) update rule on parameterized variational manifolds. First, we state and prove Theorem 4.1 to characterize the exact convergence properties of QNG updates under bounded Riemannian metrics. Crucially, we isolate the causal mechanism linking ansatz expressivity to geometric degradation, proving that the pulled-back Fubini–Study metric elements inherit the exponential O(2−N ) variance collapse dictated by Haar measure integration over random unitary circuit layouts. This scaling is shown to drive an exponential inflation of the metric condition number, establishing a clear geometric origin for the numerical tracking instabilities that stall optimization vectors in large-scale parameter spaces. Second, we formalize the Information-Geometric Quantum Brachistochrone (Theorem 4.2), proving that under instantaneous velocity limitations derived from the system’s energy variance, the absolute minimum time T required to map a state vector between two configurations is bounded strictly below by the integrated Riemannian geodesic distance Lgeodesic. This result demonstrates that flat Euclidean metrics severely underestimate physical time requirements near highly entangled boundaries where local manifold curvature dominates. Finally, we derive a sharp Shot-Noise Metric Concentration Bound (Theorem 4.3), resolving an exact cyclic trace simplification to prove that finite-shot perturbation errors concentrate according to a structural noise alignment factor κ(θ) = Tr(Σg−1)/∥∆θ∥2g . Multi-trial multinomial simulations validate this relation across all shot budgets (102 to 105), verifying that the empirical breach probability tracks safely below the analytical trace constraint line. This bound exposes a fundamental engineering trade-off: as an optimization path approaches ill conditioned regions where the minimum eigenvalue collapses (λmin → 0), the tracking framework requires a quadratic expansion of measurement resources to suppress shot noise and stabilize control loops.
Multimode squeezing is a key resource for continuous-variable quantum technologies, but its spatial range in bosonic lattices is usually tied to dispersive propagation. Here we show that parametric pairing can create an exactly flat Bogoliubov band spanned by compact Bogoliubov generators, while producing phase-sensitive anomalous correlations and sub-vacuum collective-mode squeezing that extend far beyond their finite support. Each compact generator mixes annihilation and creation operators, thereby encoding the Bogoliubov squeezing structure, and neighboring translated generators can have nonzero commutator overlap. Enforcing canonical bosonic commutation relations therefore requires spatially extended linear combinations of these translated compact generators, which define the canonical Bogoliubov modes. The squeezing transformation of these modes varies with momentum and is quantified by the squeezing-sector symplectic quantum metric. A complex-momentum singularity of the analytically continued canonical Bogoliubov modes sets both the anomalous-correlation decay length and the momentum-space width of this metric, defining an intrinsic quantum-geometric length scale. This singularity can be tuned continuously while preserving exact flatness, thereby controlling the spatial range of correlations and squeezing. Away from exact flatness, long-distance correlations persist through multiple decay channels, while modes localized by defects and dimerized boundaries provide complementary probes of the underlying quantum geometry. In the weak-damping limit, the same quantum-geometric length scale can be extracted from frequency-filtered two-port correlations. Our results establish the quantum geometry of canonical Bogoliubov modes as a mechanism for spatially extended quantum resources arising from compact Bogoliubov generators.
We employ an exact solution of the thermal bath Lindblad master equation with the Liouvillian superoperator that takes into account both dynamic and environment-induced intermode couplings to study the speed of evolution and quantum speed limit (QSL) times of a open multi-mode bosonic system. The time-dependent QSL times are defined from quantum speed limits, giving upper bounds on the rate of change of two different measures of distinguishability: the fidelity of evolution and the Hilbert-Schmidt distance. For Gaussian states, we derive explicit expressions for the evolution speed and the QSL times. General analytical results are applied to the special case of a two-mode system where the intermode couplings can be characterized by two intermode coupling vectors: the frequency vector and the relaxation rate vector. For the system initially prepared in a two-mode squeezed state, dynamical regimes are generally determined by the intermode coupling vectors, the squeezing parameter and temperature. When the vectors are parallel, different regimes may be associated with the disentanglement time, which is found to be an increasing (a decreasing) function of the length of the relaxation vector when the squeezing parameter is below (above) its temperature-dependent critical value. Alternatively, we study dynamical regimes related to the long-time asymptotic behavior of the QSL times, which is characterized by linear time dependence with the proportionality coefficients defined as the long-time asymptotic ratios. These coefficients are evaluated as a function of the squeezing parameter at varying temperatures and relaxation vector lengths. We also discuss how the magnitude and orientation of the intermode coupling vectors influence the maximum speed of evolution and dynamics of the entropy and the mutual information.