Discretized Holonomy as an Encoding Primitive for Quantum Cryptography

Abstract

Geometric phases in quantum systems give rise to holonomies associated with cyclic evolution. In mixed-state settings, these holonomies arise from Uhlmann parallel transport on the purification bundle of density operators and depend on the global geometry of density-operator trajectories. In this work we introduce discretized mixed-state holonomy as a geometric encoding primitive. The public object is a density-operator trajectory, while the encoded information resides in a finite family of purification gauges associated with the same holonomy conjugacy class. By discretizing the gauge freedom of the purification bundle, continuous mixed-state holonomy is converted into finite alphabets suitable for discrete key spaces. We formalize the resulting primitive through an operational observation model and establish several structural properties. First, the holonomy representative cannot in general be reconstructed from the density-operator trajectory alone. Second, under conjugation-invariant observation channels, different encoded symbols are indistinguishable at the level of base-space observables. Third, discretized holonomy alphabets admit stable decoding under bounded perturbations of the underlying trajectory. The construction is illustrated through explicit qubit realizations based on isospectral mixed-state loops, together with representative-level decoding mechanisms and finite holonomy alphabets. We further discuss symmetry breaking, frame leakage, and limitations arising in realistic implementations. These results isolate a primitive-level geometric encoding mechanism based on mixed-state quantum geometry and provide a structural foundation for future cryptographic constructions involving representative-level holonomy information.