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arXiv:2605.26697

A Gauge-Covariant Theoretical Framework for Non-Abelian Holonomy Estimation and Feed-Forward Correction in Time-Bin Photonic Qudits

N. Josef Bruzzese

quant-phSubmitted May 26, 2026• Announced May 27, 2026• 0 citations
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Abstract

We develop a theoretical and computational framework for estimating and correcting non-Abelian geometric distortions in time-bin photonic qudit processing when the relevant encoded object is a transported logical subspace rather than a collection of independent rays. In such settings, for example under mode mixing, multiplexed routing, or effective degeneracies, the geometric contribution is naturally matrix-valued and is described by a Wilczek-Zee holonomy on a rank-$m$ sub-bundle of the ambient Hilbert space. The framework generalises prior Abelian time-bin Pancharatnam-Berry feed-forward calibration, in which geometric distortions are represented by bin-resolved scalar phases, to the non-Abelian, matrix-valued case. We construct a gauge-covariant discrete estimator from overlap matrices between successive subspace frames: the polar factor of each overlap gives a unitary backward frame comparator, and the adjoint comparators compose to approximate the forward path-ordered exponential of the Wilczek-Zee connection. We prove gauge covariance under frame changes, polar optimality of the local comparator, consistency under partition refinement, and perturbative stability under well-conditioned overlap errors. We then formulate left- and right-acting feed-forward correction rules for removing the estimated holonomy from an effective logical operation. The work does not assume a device-specific transfer matrix, loss model, detector model, or experimental calibration pipeline; numerical studies use synthetic non-Abelian transport models to validate covariance, convergence, conditioning dependence, and correction fidelity.

Comments
21 pages, 5 figures. Reproducibility package archived at Zenodo: https://doi.org/10.5281/zenodo.19945689. Source code: https://github.com/njblottly/A-Gauge-Covariant-Theoretical-Framework-for-Non-Abelian-Holonomy-Estimation-and-Feed-Forward-Correct

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We develop a theoretical and computational framework for estimating and correcting non-Abelian geometric distortions in time-bin photonic qudit processing when the relevant encoded object is a transported logical subspace rather than a collection of independent rays. In such settings, for example under mode mixing, multiplexed routing, or effective degeneracies, the geometric contribution is naturally matrix-valued and is described by a Wilczek-Zee holonomy on a rank-mmm sub-bundle of the ambient Hilbert space. The framework generalises prior Abelian time-bin Pancharatnam-Berry feed-forward calibration, in which geometric distortions are represented by bin-resolved scalar phases, to the non-Abelian, matrix-valued case. We construct a gauge-covariant discrete estimator from overlap matrices between successive subspace frames: the polar factor of each overlap gives a unitary backward frame comparator, and the adjoint comparators compose to approximate the forward path-ordered exponential of the Wilczek-Zee connection. We prove gauge covariance under frame changes, polar optimality of the local comparator, consistency under partition refinement, and perturbative stability under well-conditioned overlap errors. We then formulate left- and right-acting feed-forward correction rules for removing the estimated holonomy from an effective logical operation. The work does not assume a device-specific transfer matrix, loss model, detector model, or experimental calibration pipeline; numerical studies use synthetic non-Abelian transport models to validate covariance, convergence, conditioning dependence, and correction fidelity.
We develop a geometric-phase framework for time-bin photonic qudits and propose a practical calibration and feed-forward algorithm for separating and compensating geometric (Pancharatnam-Berry), dynamical, and technical phase contributions. Working directly in the time-bin basis, we use a parallel-transport gauge so that geometric phases appear as experimentally identifiable interferometric offsets, while all phase contributions enter a bin-resolved diagonal transformation. We model state preparation by cascaded unbalanced Mach-Zehnder interferometers and give closed-form amplitudes for arbitrary splitting ratios and phases, noting that single-port monitoring requires post-selection and renormalization. We then give an interferometric tomography recipe based on adjacent-bin scans, with a Fourier-basis cross-check, and a multi-mode numerical case study that separates total, dynamical, and geometric phases and demonstrates feed-forward compensation. The protocol uses standard components, including tunable UMZIs, phase shifters or EOMs, and single-photon detectors, together with routine phase sweeps. It is intended for small to moderate dimensions, approximately d up to 10, and provides a scalable route toward phase-stable high-dimensional temporal encoding for quantum communication and photonic processing.
Abstract: This work develops a unified and gauge-invariant framework in which quantum interference is described as a measurement of relative holonomy between two coherent evolutions in a combined configuration space X = M × Λ. Both the Aharonov–Bohm (Abelian) and Wilczek–Zee (non-Abelian) effects appear as limits of a single Wilson-loop observable. A block connection is introduced on the direct-sum bundle E = H ⊕ (T X ⊗ ℂ), with structure group U(n) × O(p,q), thereby unifying Berry parallel transport and Levi-Civita geometry within one connection. The formalism ensures gauge and diffeomorphism covariance, analytic well-posedness under back-reaction, and establishes conditions for reconstructing the spacetime metric from holonomy data. The resulting theory provides a mathematically consistent and experimentally testable synthesis of geometric phase, interference, and spacetime connection. Potential physical implementations include mesoscopic rings, cold-atom systems, and superconducting circuits. Keywords: geometric phase; holonomy; Aharonov–Bohm effect; Berry phase; Wilczek–Zee connection; differential geometry; quantum interference; gauge theory.
Geometric phases encode the holonomy acquired under cyclic control of a quantum Hamiltonian: for nondegenerate eigenstates the effect is Abelian (a scalar phase), while in degenerate manifolds the Wilczek-Zee generalization yields a matrix-valued holonomy that can mix states and depends on path ordering. We present an overview of measuring Abelian and non-Abelian geometric phases in nitrogen-vacancy (NV) centers in diamond and propose a non-rotating measurement protocol enabled by the intrinsic14 N{ }^{14} ~\mathrm{N}14 Nhyperfine manifold. As a first step, we propose to measure the Abelian Berry phase on a single hyperfine-resolved two-level transition using microwaveengineered closed loops with dynamical-phase cancellation by loop reversal. We then propose a non-Abelian protocol that engineers a two-dimensional degenerate dark subspace in an NV-14 N{ }^{14} ~\mathrm{N}14 Nmanifold via RF dressing and three phase-coherent microwave couplings (a tripod Hamiltonian). Two closed control loops implemented by winding independent microwave phases generate non-commuting SU(2) holonomies, which can be verified using final-only population readout and minimal tomography. We also present timing trade-off curves, full unitary Schrödinger evolution of the four-level tripod, and projected dark-subspace diagnostics that quantify both leakage and non-commuting holonomy.
Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov--Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C){\rm GL}(N,\mathbb C)GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.
Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here, we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov-Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.
Stokes' theorem turns Abelian Berry phases into curvature fluxes, whereas path ordering precludes such a simple formula for non-Abelian holonomies. We show that a quantitative form of this intuition survives: arbitrary Wilczek--Zee holonomies obey a universal quantum geometric limit~(QGL), in which the holonomy magnitude is bounded by a surface integral of the non-Abelian curvature norm. Recasting holonomic evolution as an effective Stokes--Schrödinger dynamics driven by transported curvature, we identify the QGL as the geometric counterpart of conventional quantum speed limits, with a time-integrated generator norm replaced by a surface-integrated curvature cost. The induced contour--surface variational problem is governed by a non-Abelian Lorentz force, which we address with a brachistochrone ansatz of curvature-weighted geodesics. Applied to an SU(2) tripod dark subspace, near-optimal protocols spontaneously align the transported curvature along a single Lie-algebra direction, effectively taming non-Abelianity.