We develop a unified geometric framework in which the quantum geometric tensor (QGT) simultaneously governs both the metrology of gauge-invariant curvature and the symmetry classification of random-matrix ensembles. The symmetric Fubini–Study sector, generated by the Jordan product of symmetric logarithmic derivatives (SLDs), determines the quantum Fisher information and therefore the information gain associated with closed-loop curvature acquisition. The antisymmetric Berry sector, classified by the de Rham cohomology class of the SLD connection, determines whether the accumulated holonomy phase is linearly observable. We show that time-reversal symmetry acts as a topological gate on the readable Berry channel: the holonomy survives only in the unitary Wigner–Dyson class (GUE), where the cohomology class is nontrivial. This converts the previously phenomenological time-reversal “gate factor” into a cohomological indicator. In the symplectic class (GSE), the non-Abelian obstruction is identified with the Jordan associator curvature and appears operationally as a Holevo incompatibility penalty. The resulting cost-efficiency functional is shown to be genuinely U-shaped across the Dyson classes, with a strict minimum at GUE arising from the non-monotonic topology of the readable holonomy sector. We further establish a bridge between the QGT and thermodynamic fluctuation geometry, derive curvature-statistics scaling laws from Wigner level repulsion, and analyze the deformation of the Kähler structure under decoherence through the Bures metric and mean Uhlmann curvature. The framework unifies Berry geometry, quantum metrology, random-matrix universality, and information geometry into a single cohomological description of curvature acquisition.
We develop a unified geometric framework in which the quantum geometric tensor (QGT) simultaneously governs both the metrology of gauge-invariant curvature and the symmetry classification of random-matrix ensembles. The symmetric Fubini–Study sector, generated by the Jordan product of symmetric logarithmic derivatives (SLDs), determines the quantum Fisher information and therefore the information gain associated with closed-loop curvature acquisition. The antisymmetric Berry sector, classified by the de Rham cohomology class of the SLD connection, determines whether the accumulated holonomy phase is linearly observable. We show that time-reversal symmetry acts as a topological gate on the readable Berry channel: the holonomy survives only in the unitary Wigner–Dyson class (GUE), where the cohomology class is nontrivial. This converts the previously phenomenological time-reversal “gate factor” into a cohomological indicator. In the symplectic class (GSE), the non-Abelian obstruction is identified with the Jordan associator curvature and appears operationally as a Holevo incompatibility penalty. The resulting cost-efficiency functional is shown to be genuinely U-shaped across the Dyson classes, with a strict minimum at GUE arising from the non-monotonic topology of the readable holonomy sector. We further establish a bridge between the QGT and thermodynamic fluctuation geometry, derive curvature-statistics scaling laws from Wigner level repulsion, and analyze the deformation of the Kähler structure under decoherence through the Bures metric and mean Uhlmann curvature. The framework unifies Berry geometry, quantum metrology, random-matrix universality, and information geometry into a single cohomological description of curvature acquisition.
We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum of the QFIM is invariant under unitary dynamics generated by these same operators. Each Lie algebra therefore endows any quantum state with a fixed "budget" of metrological sensitivity -- an intrinsic resource that we show, like optical squeezing in interferometry, cannot be amplified by symmetry-preserving operations. The Uhlmann curvature tensor (UCT) naturally inherits the same symmetry group, and so quantum incompatibility is similarly fixed. As a result, a metrological analog to Liouville's theorem appears; statistical distances, volumes, and curvatures are invariant under the evolution generated by the Lie algebra. We discuss this as it relates to the quantum analogs of classical optimality criteria. This enables one to efficiently classify useful classes of quantum states at the level of Lie algebras through geometric invariants.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2, with εμ denoting the parameter displacement from the EP along an individual parameter axis μ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
This paper establishes a geometric bridge between two operational results concerning gravity-mediated entanglement and phase-based field structure. First, prior work showed that entanglement in minimal gravity-mediated interferometric models arises if and only if the interaction phases contain a non-factorable plaquette invariant. This invariant represents an oriented loop phase that cannot be decomposed into local contributions and therefore serves as a necessary and sufficient condition for entanglement generation. Second, the Harmonic Code framework describes persistent physical effects as loop-structured phase geometry, in which closed phase loops encode enduring information about curvature and field structure. The present work unifies these viewpoints by demonstrating that the plaquette phase emerges naturally as a curvature-induced loop holonomy. An effective phase-gradient field is introduced, and the oriented loop phase is defined as Π = ∮ k · dℓ = ∬ (∇ × k) · dA, showing that the non-factorable phase invariant is directly generated by the curl of the phase-gradient field. A minimal rotational field model is used to derive the relation Π = 2εA, where A is the enclosed area and ε is the curl parameter. Numerical simulations verify the key geometric properties expected of a true holonomy: Linear scaling with enclosed area Shape independence at fixed area Sign reversal under orientation flip Convergence under discretization refinement The loop phase Π is then mapped to standard two-qubit entanglement measures. For the separable input state |++⟩, the concurrence and negativity are shown to depend only on Π: C(Π) = |sin(Π/2)| N(Π) = C/2 These results demonstrate that the non-factorable phase structure required for gravity-mediated entanglement has a direct geometric origin as a curvature-induced loop holonomy. The paper therefore provides a unified operational statement: Curvature generates loop phases, and those loop phases are exactly the structures responsible for gravity-mediated entanglement. This establishes a concrete geometric mechanism linking spacetime curvature, phase holonomy, and quantum entanglement, and offers a clear, experimentally testable framework for probing quantum aspects of gravity.
We develop a response-geometric framework for open quantum systems by decomposing the stationary-state response tensor into symmetric and antisymmetric sectors. The symmetric sector defines a metric-like response tensor governing local susceptibility, while the antisymmetric sector defines a curvature two-form associated with nonreciprocal response and geometric work. We show that the work curvature is precisely the antisymmetric component of the response tensor, establishing a fluctuation--response relation that extends the geometric structure of equilibrium thermodynamics to nonequilibrium steady states, and reveals a response geometry with both metric and symplectic sectors. In equilibrium, the antisymmetric sector vanishes by reciprocity, recovering the familiar metric geometry of thermodynamic response. Open quantum systems admit a broader structure in which reciprocal and nonreciprocal responses coexist on the same control manifold. We illustrate this with a driven dissipative qubit under pure dephasing, where finite curvature emerges from the misalignment between the Hamiltonian eigenbasis and the pointer basis selected by the environment, without requiring strong driving or engineered reservoirs. Comparison with the Bures metric shows that response geometry and information geometry characterize distinct properties of the stationary-state manifold: the former governs susceptibility and geometric work, the latter quantifies state distinguishability. Geometric work thus emerges as a measurable signature of nonreciprocal response in open quantum systems.
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 preprint presents a unified holonomy-first, invariant-only simulator architecture for loop-based quantum operations, integrating two complementary components:(i) an operational framework for discovering, diagnosing, and compiling native holonomic operations from closed-loop executions, and(ii) a minimal semantic language that characterizes what such loop-based operations mean when only conjugacy-invariant data are accessible. The simulator treats holonomy words—compositions of calibrated loop primitives—as the fundamental operational objects, rather than Hamiltonians, generators, or gauge-fixed operators. All semantics are defined at the level of conjugacy classes, represented operationally by invariant fingerprints such as trace moments or overlap-derived scalars. This invariant-first stance enables robust workflows that avoid full operator reconstruction, tomography, or fixed-basis assumptions. On this substrate, the simulator introduces a geometric compilation pipeline that: Extracts interaction structure from short loop compositions using commutator and nested-commutator diagnostics; Organizes noncommutativity and curvature information into a gauge-invariant interaction spectrum; Promotes reproducible, scale-consistent loop words into reusable native operations. A central contribution of the unified framework is a theorem-level semantic obstruction: the default single-system semantic tier is generically incomplete in the presence of entanglement-sensitive loop effects. This is demonstrated by introducing entanglement holonomy, a word-level invariant accessible via joint return amplitudes of entangled probes yet invisible to any reconstruction based solely on single-system class functions or mode-preserving data. As a consequence, the simulator treats the effective representation on which holonomies act as provisional rather than fixed. When invariant closure and scaling tests fail at a given tier, the simulator is forced to lift representation (e.g., from single-particle to multi-particle or encoded subspaces) as an operational necessity rather than a modeling choice. An explicit minimal example of an irreducible two-particle holonomic gate is provided within a quadratic-mode backend. The simulator does not attempt to unify classical and quantum dynamics or to provide a theory of quantum gravity. Instead, it supplies a shared holonomy-based semantic layer in which classical geometric transport and genuinely quantum effects (including entanglement) can be jointly represented and operationally distinguished using the same invariant diagnostics. This unified manuscript synthesizes and extends two previously developed components—an operational framework (GQF, DOI: 10.5281/zenodo.18303288) and a semantic foundation with an obstruction result (MHQL, DOI: 10.5281/zenodo.18332083)—to expose a structural point not apparent when they are treated separately: invariant-first compilation and representation lifting are linked by a principled semantic incompleteness theorem. The result is a rare simulator architecture that is holonomy-driven, gauge-invariant by construction, and representation-adaptive when compelled by invariant evidence.