We define a basis-independent two-particle return-amplitude residue H_E = sin²(2γ) sin⁴(χ/2), constructed from a single-particle return angle χ ∈ [0, π] and a local dyad rotation γ. The residue is bounded in [0, 1], scales quartically for small χ, and isolates an off-diagonal cross term that appears in the symmetric entangled return amplitude but not in the corresponding product-state amplitude. It is invariant under common unitary changes of basis at the basepoint and depends only on operational overlap data, not on the mechanism that produces the loop. A minimal number-conserving two-boson hopping Hamiltonian is then shown to carry a nontrivial encoded holonomic gate. The encoded subspace span{|20⟩, |02⟩} satisfies the scalar-compression condition with vanishing compressed Hamiltonian, and at T = π/(2J) the cyclic return map is U_code = −σ_x, purely geometric in the projected-evolution sense. The associated one-particle obstruction is calibrated using higher-rank numerical ranges. For a Hermitian one-body Hamiltonian κ on m modes, the rank-2 scalar-compression condition is governed by Λ_2(κ) = [λ_2, λ_{m−1}]. The obstruction holds at m = 2 unless κ is scalar, collapses to a single value at m = 3, and becomes a generic interval at m ≥ 4. The two-boson construction is therefore a minimal mode-resource separation, not an absolute single-particle no-go theorem. The paper also gives an interferometric estimation protocol for H_E, the resulting shot-count scaling, and a comparison with Lusk's spacetime-specific entanglement holonomy in Kerr geometry.
We define a basis-independent two-particle return-amplitude residue H_E = sin²(2γ) sin⁴(χ/2), constructed from a single-particle return angle χ ∈ [0, π] and a local dyad rotation γ. The residue is bounded in [0, 1], scales quartically for small χ, and isolates an off-diagonal cross term that appears in the symmetric entangled return amplitude but not in the corresponding product-state amplitude. It is invariant under common unitary changes of basis at the basepoint and depends only on operational overlap data, not on the mechanism that produces the loop. A minimal number-conserving two-boson hopping Hamiltonian is then shown to carry a nontrivial encoded holonomic gate. The encoded subspace span{|20⟩, |02⟩} satisfies the scalar-compression condition with vanishing compressed Hamiltonian, and at T = π/(2J) the cyclic return map is U_code = −σ_x, purely geometric in the projected-evolution sense. The associated one-particle obstruction is calibrated using higher-rank numerical ranges. For a Hermitian one-body Hamiltonian κ on m modes, the rank-2 scalar-compression condition is governed by Λ_2(κ) = [λ_2, λ_{m−1}]. The obstruction holds at m = 2 unless κ is scalar, collapses to a single value at m = 3, and becomes a generic interval at m ≥ 4. The two-boson construction is therefore a minimal mode-resource separation, not an absolute single-particle no-go theorem. The paper also gives an interferometric estimation protocol for H_E, the resulting shot-count scaling, and a comparison with Lusk's spacetime-specific entanglement holonomy in Kerr geometry.
Holonomy-first quantum frameworks treat closed-loop transformations — holonomies — as the primitive operations from which native gates are mined, rather than as derived quantities. We develop this viewpoint in two parts. First, we construct a gauge-invariant commutator-strength matrix S(ε) from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of S(ε) is asymptotic with a general O(ε) remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving O(ε2) accuracy, and a counterexample shows the symmetry cannot be dropped. The promotion criteria are operationalized as declared statistical tests — sampling-cost laws for interferometric and readout-floor regimes, a bias-corrected scale-equivalence rule, and multiplicity control — and a deterministic simulation campaign reports decision replication, false-promotion control, inverse-defect detection, and execution cost under finite-shot readout, calibration error, imperfect inverses, and visibility loss. Second, for two-particle preparations of Schmidt rank two, the common-basepoint return-amplitude residue factorizes exactly as ΔPθ = C(θ)HE(χ, γ), with C(θ) the concurrence and HE = sin2(2γ) sin4(χ/2) a calibrated loop factor, determining negativity, entanglement entropy, product fidelity, and geometric entanglement within the family. A two-boson construction realizes the encoded geometric gate −σx on span{|20⟩, |02⟩}; under the same two-mode, same-dimension projected-scalar condition, no corresponding one-particle holonomic cycle exists unless the one-body Hamiltonian is scalar. The mathematical ingredients are established; the contribution is the invariant-readout protocol, its quantified cost and uncertainty analysis, and precise statements of what can and cannot be inferred without full operator reconstruction. This record contains the abstract only. It is not the full text of the article. The complete peer-reviewed article is published open access in Discover Quantum Science and is freely available from the publisher at https://doi.org/10.1007/s44464-026-00037-5 — including all figures, tables, appendices and the supplementary code and data. Cite as: Kazadi J. Holonomy-first native operations geometric quantum framework and entanglement-sensitive multi-particle holonomic gates. Discover Quantum Science. 2026;2:35. https://doi.org/10.1007/s44464-026-00037-5
Abstract Holonomy-first quantum frameworks treat closed-loop transformations—holonomies—as the primitive operations from which native gates are mined, rather than as derived quantities. We develop this viewpoint in two parts. First, we construct a gauge-invariant commutator-strength matrix S(ϵ) from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of S(ϵ) is asymptotic with a general O(ϵ) remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving O(ϵ2) accuracy, and a counterexample shows the symmetry cannot be dropped. The promotion criteria are operationalized as declared statistical tests—sampling-cost laws for interferometric and readout-floor regimes, a bias-corrected scale-equivalence rule, and multiplicity control—and a deterministic simulation campaign reports decision replication, false-promotion control, inverse-defect detection, and execution cost under finite-shot readout, calibration error, imperfect inverses, and visibility loss. Second, for two-particle preparations of Schmidt rank two, the common-basepoint return-amplitude residue factorizes exactly as ΔPθ=C(θ)HE(χ,γ) , with C(θ) the concurrence and HE=sin2(2γ)sin4(χ/2) a calibrated loop factor, determining negativity, entanglement entropy, product fidelity, and geometric entanglement within the family. A two-boson construction realizes the encoded geometric gate −σx on span{∣20⟩,∣02⟩} ; under the same two-mode, same-dimension projected-scalar condition, no corresponding one-particle holonomic cycle exists unless the one-body Hamiltonian is scalar. The mathematical ingredients are established; the contribution is the invariant-readout protocol, its quantified cost and uncertainty analysis, and precise statements of what can and cannot be inferred without full operator reconstruction.
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.
This preprint develops a state-level framework in which holonomy is treated as a property of cyclic subspaces rather than single-particle eigenstates or gauge fields. Within number-conserving quadratic Hamiltonian dynamics on bosonic or fermionic modes, a precise notion of single-particle irreducibility is introduced, meaning that no compatible single-particle cyclic subspace of the same dimension can satisfy a holonomic (purely geometric) condition. The main result is an explicit existence theorem showing that nontrivial, purely geometric holonomies can arise in two-particle cyclic subspaces even when the associated single-particle problem admits no holonomic cyclic subspace of the same dimension. A fully worked minimal example—two bosons in two modes—is provided, yielding a nontrivial SU(2) holonomy on an encoded two-dimensional subspace while the single-particle sector is provably non-holonomic. The paper is self-contained and includes a conservative, device-agnostic certification scheme based on loop composition and conjugacy-invariant readouts, enabling experimental verification without model-specific gauge-field assumptions.
A spectrally isolated quartet can admit a local two-qubit description at each point in parameter space and still acquire a loop holonomy outside the local subgroup U(2)⊗U(2). We study this question in three localized topological settings, a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet. On a fixed quartet, changing only the loop can move the holonomy from almost local to entangling. In BHZ, co-rotating and counter-rotating edge-field loops have nearly the same eigenphase data, but only the counter-rotating loop yields an Ising-like entangler. SSH gives a controlled-rotation example in a numerically stable edge quartet. BBH shows the same issue in a higher-order corner quartet. Standard Berry data, including Berry phases, Chern numbers, determinant phases, and eigenphase spectra, do not separate these cases. The main diagnostic is the distance from the loop holonomy to the extracted local subgroup. Canonical two-qubit coordinates are used only after reduction failure has been identified. The quartet is the smallest setting in which this question can be tested explicitly. The same subgroup-reduction problem extends to any isolated multiplet with pointwise product type D=∏αdα, where the relevant local subgroup is the embedded product group Gd=Im[∏αU(dα)→U(D)]. In the terminology of Ref.[arXiv:2601.13764], these examples realize loop-dependent entangling gluing.
Abstract We develop a fully covariant, operational link between general relativity and quantum measurement based on reflective holonomy : the spatial rotation induced in an apparatus rest frame when its tetrad is transported around a spacetime loop and projected onto the stabilizer of the apparatus four-velocity. This rotation determines the realized measurement basis in interferometric and Bell-type experiments. Shot-to-shot fluctuations in reflective holonomy—arising from spacetime curvature, platform dynamics, or control jitter—produce random-unitary basis noise that suppresses Ramsey visibility and Bell–CHSH values when not recorded, yet is fully reversible through per-shot logging and inversion. The framework requires no modification of either general relativity or quantum mechanics and yields a falsifiable prediction: complete logging and inversion must restore quantum-limited performance, with any residual deviation signaling non-holonomic physics. From first principles we derive (i) the holonomy-to-quantum-channel map, (ii) its Lindblad limit characterized by a dephasing rate proportional to the variance growth of the holonomy, (iii) closed-form expressions for Ramsey and CHSH contrasts, and (iv) a minimal small-loop variance law linking curvature to basis noise. Analytical and numerical studies show that several longstanding experimental anomalies—including anomalously low CHSH values exhibiting the characteristic three-damped–one-stable correlator pattern, and Ramsey contrast losses exceeding intrinsic decoherence—are precisely accounted for by untracked holonomy fluctuations and vanish under inversion, while no-signalling and Tsirelson’s bound remain preserved. We formulate a one-parameter Operational Unification Test : determine the holonomy-noise parameter from Ramsey data, predict CHSH values with no additional parameters, and confirm full recovery after inversion. Irreducible residuals isolate physics that cannot be attributed to reflective holonomy.