Date of publication
Oct 22, 2026
Loop-dependent entangling holonomies in localized topological quartets
by
Kazuki Ikeda
,
Yaron Oz
Non-Abelian adiabatic holonomies are used both to diagnose topology and to implement geometric quantum operations. Yet the standard descriptors of such transport --- Berry phases, Chern numbers, determinant phases, and Wilson-loop spectra --- do not determine whether the resulting holonomy acts as local logical operations on an extracted subsystem structure. We identify this missing information as a subgroup-reduction problem. For an isolated multiplet with pointwise product type $D=\prod _{\alpha } d_{\alpha }$, we ask whether the loop holonomy lies in the embedded local subgroup $G_{\mathbf d}=\operatorname {Im}[\prod _{\alpha }\mathrm {U}(d_{\alpha })\to \mathrm {U}(D)]$. We show that a multiplet can remain pointwise local throughout parameter space while different loops on the same control manifold realize sharply different logical actions, from almost local transport to controlled operations and entangling holonomies, even when the usual Berry or Wilson-loop data are nearly unchanged. The primary diagnostic is the distance of the microscopic holonomy to the extracted local subgroup, relative to a specified product reduction; gate-specific invariants are invoked only after reduction failure has been established. This identifies a quantum-geometric observable that is invisible to standard spectral reductions and directly motivates tomography-based detection of hidden subsystem-nonlocality. Explicit localized topological benchmarks are provided by a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet.