Squash Operator and Symmetry

This paper begins with a simple proof of the existence of squash operators compatible with the Bennett-Brassard 1984 (BB84) protocol which suits single-mode as well as multi-mode threshold detectors. The proof shows that, when a given detector is symmetric under cyclic group C_4, and a certain observable associated with it has rank two as a matrix, then there always exists a corresponding squash operator. Next, we go on to investigate whether the above restriction of "rank two" can be eliminated; i.e., is cyclic symmetry alone sufficient to guarantee the existence of a squash operator? The motivation behind this question is that, if this were true, it would imply that one could realize a device-independent and unconditionally secure quantum key distribution protocol. However, the answer turns out to be negative, and moreover, one can instead prove a no-go theorem that any symmetry is, by itself, insufficient to guarantee the existence of a squash operator.