Revisiting optimal eavesdropping in quantum cryptography: Optimal interaction is unique up to rotation of the underlying basis

A general framework of optimal eavesdropping on BB84 protocol was provided by Fuchs et al. [Phys. Rev. A 56, 1163 (1997)]. An upper bound on mutual information was derived, which could be achieved by a specific type of interaction and the corresponding measurement. However, uniqueness of optimal interaction was posed as an unsolved problem there and it has remained open for almost two decades now. In this paper, we solve this open problem and establish the uniqueness of optimal interaction up to rotation. The specific choice of optimal interaction by Fuchs et al. is shown to be a special case of the form derived in our work.