Squashing models for optical measurements in quantum communication

Measurements with photodetectors are naturally described in the infinite dimensional Fock space of one or several modes. For some measurements, a model has been postulated which describes the full 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 full and target measurements can be connected by a squashing model. We show that a measurement used in the Bennett-Brassard 1984 (131384) protocol does allow a squashing description, although the corresponding six-state protocol measurement 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.