Squashing Models for Optical Measurements in Quantum Communication

Abstract

Measurements with photodetectors necessarily need to be described in the infinite dimensional Fock space of one or several modes. For some measurements a model has been postulated which describes the full mode measurement as a composition of a mapping (squashing) of the signal into a small dimensional Hilbert space followed by a specified target measurement. We present a formalism to investigate whether a given measurement pair of mode and target measurements can be connected by a squashing model. We show that the measurements used in the BB84 protocol do allow a squashing description, although the six-state protocol does not. As a result, security proofs for the BB84 protocol can be based on the assumption that the eavesdropper forwards at most one photon, while the same does not hold for the six-state protocol.