Counterfactual quantum cryptography

Abstract

Counterfactual Quantum Cryptography is a counter-intuitive scheme where two remote parties can share a secret key without the transmission of a photon between them, hence the term "counterfactual". This concept is fundamentally based on the probability of transmission. When a discrete weak measurement is done on the system by an external party, the transmission channel's wave distribution evolves by a small translation which leaves a phase error in the system, thereby affecting the counterfactual probability. In this article, we illustrate how the optical field changes with accordance to the errors in the system. Consequently, we evaluate the counterfactual probability of distributing a key when a weak measurement is done and compare it to an error in the source. We prove that the information Eve obtains, lacks usefulness and suggest ways to distinguish the two errors in the protocol which includes increasing the angular frequency while applying a small change in phase of the Beam Splitter.