Mathematical methods of quantum cryptography

Quantum cryptography is a modern branch of science where methods of secure communication based on principles of quantum mechanics are studied. The rigorous proof of the security of quantum cryptography (more precisely, quantum key distribution) gave rise to a complex and beautiful mathematical theory, which is based on methods of quantum information theory, in particular, quantum entropic measures and entropic uncertainty relations. In the talk, well-known results in the proof of security of quantum key distribution will be reviewed. Currently, an important task is to prove the security of quantum key distribution with imperfect devices. This is important for practical implementations. In papers [1, 2], theorems on the security of the famous BB84 quantum key distribution protocol (the first and most widely used protocol) with detection-efficiency mismatch were rigorously proved for the first time. The differences in the detector efficiencies (the probabilities of detecting a photon) destroys certain symmetries in the problem, which makes the earlier methods of security proofs insufficient. In [1, 2], new methods for analytic solution of a minimization problem of a certain convex functional (quantum relative entropy) and methods for bounding the dimensionality of the search space under were developed. This was one of the key problems for solving this problem. In [3], a number of statements were proved that explain the operational meaning of the security parameter widely used in quantum key distribution.

References

1. M. K. Bochkov, A. S. Trushechkin, “Security of quantum key distribution with detection-efficiency mismatch in the single-photon case: Tight bounds”, Phys. Rev. A, 99:3 (2019), 32308, 15 pp.        
2. A. S. Trushechkin, “Security of quantum key distribution with detection-efficiency mismatch in the multiphoton case”, 2020, 18 pp., arXiv: 2004.07809
3. A. S. Trushechkin, “On the operational meaning and practical aspects of using the security parameter in quantum key distribution”, Quantum Electron., 50:5 (2020), 426–439