Upper bounds on the leakage of private data and operational approach to markovianity

Abstract

We quantify consequences of a leakage of the private key and private randomness generated during quantum key distribution. We provide simple lower bounds on the one-way distillable key after the leakage has been detected. We also show that the distributed private randomness does not drop down by more than twice the number of qubits of the traced-out system. We further focus on strictly irreducible private states, showing that their two-way distillable key is to same extent non-lockable. We achieve this by referring to the idea of recovery maps. We then consider special case of side-channels on some of private states. With the view of usage of the latter states in the hybrid quantum network we consider private state construction based on quantum secrete sharing and show its limitations. Finally, in analogy to a recent result of J. Ko lodyński et al. [Phys. Rev. A 101, 020303(R) (2020)], we connect the topic of (non)markovian dynamics with that of hacking. In particular, we show that an invertible map is non-CP-divisible if and only if there exists a state whose the key witness by a particular privacy witness increases in time.