Indefinite causal key distribution

Abstract

We propose a quantum key distribution (QKD) protocol that is carried out in an indefinite causal order (ICO). In QKD, one considers a setup in which two parties, Alice and Bob, share a key with one another in such a way that they can detect whether an eavesdropper, Eve, has learnt anything about the key. To our knowledge, in all QKD protocols proposed until now, Eve is detected by publicly comparing a subset of Alice and Bob's key and checking for errors. We find that a consequence of our protocol is that it is possible to detect eavesdroppers without publicly comparing any information about the key. Indeed, we prove that it is not possible for eavesdroppers, performing any individual attack, to extract useful information about the shared key without inducing a nonzero probability of being detected. We also prove the security of this protocol against a class of individual eavesdropping attacks. The role ICO plays in causing unusual phenomena in quantum technologies is an important question. By considering it we find a two-way QKD protocol that exhibits a similar private detection feature, albeit with some interesting differences. After noting some implications of these differences and discussing some of the practicalities of our protocol, we conclude that this work is best considered as a first step in applying quantum cryptographic ideas in an ICO.