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 suit... More >

Mixed quantum states exhibit a rich geometric structure from the information geometry of density operators and their purifications. Parallel transport via the Uhlmann connection induces holonomy transformations along closed trajectories of density operators. Although extensively studied in quantum physics, their potential as information-theoretic resources in quantum cryptography remains largely unexplored. This work investigates whether holonomy in the purification bundle can serve as a secrecy resource for mixed-state quantum systems. The key observation is that while a density-operator trajectory describes only the base manifold, the lifted trajectory depends on an additional unitary gaug... More >