Squash Operator and Symmetry

TL;DR

This paper proves the existence of squash operators under certain symmetry conditions in quantum key distribution and shows that symmetry alone cannot guarantee their existence, establishing a no-go theorem.

Contribution

It provides a simple proof of squash operators for symmetric detectors and demonstrates that symmetry alone is insufficient for their existence, leading to a no-go theorem.

Findings

Existence of squash operators for cyclic symmetric detectors with rank-two observables

Symmetry alone does not guarantee squash operator existence

No-go theorem for symmetry-based guarantees of squash operators

Abstract

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.