Symmetries in Quantum Key Distribution and the Connection between Optimal Attacks and Optimal Cloning

TL;DR

This paper explores the relationship between optimal eavesdropping attacks and optimal cloning in quantum key distribution, revealing that the connection depends on protocol symmetries and the number of mutually unbiased bases used.

Contribution

It establishes conditions under which optimal attacks correspond to optimal cloners in symmetric QKD protocols, and identifies classes where this correspondence holds or fails.

Findings

Optimal attack equals optimal cloner for protocols with 2 and d+1 MUBs.

For protocols with d MUBs, optimal attack is not an optimal cloner.

Criteria are provided to identify protocols with identical optimal attacks despite different states.

Abstract

We investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack where the eavesdropper employs an optimal cloner to attack the quantum key distribution (QKD) protocol. The analysis is done in the context of the security proof in [Devetak and Winter, Proc. of the Roy. Soc. of London Series A, 461, 207 (2005); Kraus, Gisin and Renner, Phys. Rev. Lett. 95, 080501 (2005)] for discrete variable protocols in d-dimensional Hilbert spaces. We consider a scenario in which the protocols and cloners are equipped with symmetries. These symmetries are used to define a quantum cloning scenario. We find that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, where we can identify an optimal attack by an optimal cloner. We analyze protocols with 2, d and d+1 mutually unbiased bases where d is a prime, and show that for the protocols with 2 and d+1 MUBs the optimal attack is an optimal cloner, but for the protocols with d MUBs, it is not. Finally, we give criteria to identify protocols which have different signal states, but the same optimal attack. Using these criteria, we present qubit protocols which have the same optimal attack as the BB84 protocol or the 6-state protocol.