TY  - JOUR
AU  - Morsalani, Mohamed El
PY  - 2026
DA  - 2026/05/17
TI  - Secret Holonomy and Public Geometry in Mixed-State Quantum Information
JO  - Journal of Quantum Cryptography
T2  - Journal of Quantum Cryptography
JF  - Journal of Quantum Cryptography
VL  - 1
IS  - 1
SP  - 35
EP  - 54
DO  - 10.62762/JQC.2026.133798
UR  - https://www.icck.org/article/abs/JQC.2026.133798
KW  - mixed-state quantum cryptography
KW  - uhlmann holonomy
KW  - quantum information geometry
KW  - bures geometry
KW  - purification bundle
KW  - geometric secrecy
AB  - 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 gauge freedom associated with purification amplitudes. Consequently, distinct lifted trajectories from different gauge choices may generate different holonomy transformations even when the observable density evolution remains identical. We formalize this by proving that holonomy from a closed density-operator trajectory cannot be uniquely reconstructed from base-space observations. Specifically, the observable trajectory determines holonomy only up to conjugacy in the unitary group. This reveals a structural separation between publicly observable geometric information—namely the density-operator trajectory equipped with Bures quantum information geometry—and hidden bundle-level transformations arising from purification gauge choices. Based on this property, we introduce a minimal geometric encoding primitive where classical symbols correspond to gauge transformations in the purification bundle while the observable density trajectory remains unchanged. We further formalize a base-space-restricted adversarial model, define a distinguishability advantage for message recovery, and analyze potential attacks including tomographic, geometric, and algebraic reconstruction attempts. Explicit single-qubit and multi-qubit examples illustrate how Abelian and non-Abelian holonomy arise in low-dimensional mixed-state systems. These results identify holonomy as a geometric information carrier in mixed-state quantum information and provide structural foundations for future holonomy-based quantum cryptographic protocols.
SN  - request pending
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Morsalani2026Secret,
  author = {Mohamed El Morsalani},
  title = {Secret Holonomy and Public Geometry in Mixed-State Quantum Information},
  journal = {Journal of Quantum Cryptography},
  year = {2026},
  volume = {1},
  number = {1},
  pages = {35-54},
  doi = {10.62762/JQC.2026.133798},
  url = {https://www.icck.org/article/abs/JQC.2026.133798},
  abstract = {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 gauge freedom associated with purification amplitudes. Consequently, distinct lifted trajectories from different gauge choices may generate different holonomy transformations even when the observable density evolution remains identical. We formalize this by proving that holonomy from a closed density-operator trajectory cannot be uniquely reconstructed from base-space observations. Specifically, the observable trajectory determines holonomy only up to conjugacy in the unitary group. This reveals a structural separation between publicly observable geometric information—namely the density-operator trajectory equipped with Bures quantum information geometry—and hidden bundle-level transformations arising from purification gauge choices. Based on this property, we introduce a minimal geometric encoding primitive where classical symbols correspond to gauge transformations in the purification bundle while the observable density trajectory remains unchanged. We further formalize a base-space-restricted adversarial model, define a distinguishability advantage for message recovery, and analyze potential attacks including tomographic, geometric, and algebraic reconstruction attempts. Explicit single-qubit and multi-qubit examples illustrate how Abelian and non-Abelian holonomy arise in low-dimensional mixed-state systems. These results identify holonomy as a geometric information carrier in mixed-state quantum information and provide structural foundations for future holonomy-based quantum cryptographic protocols.},
  keywords = {mixed-state quantum cryptography, uhlmann holonomy, quantum information geometry, bures geometry, purification bundle, geometric secrecy},
  issn = {request pending},
  publisher = {Institute of Central Computation and Knowledge}
}