Covariance Matrices under Bell-like Detections

Abstract

We derive a simple formula for the transformation of an arbitrary covariance matrix of (n + 2) bosonic modes under general Bell-like detections, where the last two modes are combined in an arbitrary beam splitter (i.e., with arbitrary transmissivity) and then homodyned. In particular, we consider the realistic condition of non-unit quantum efficiency for the homodyne detectors. This formula can easily be specialized to describe the standard Bell measurement and the heterodyne detection, which are exploited in many contexts, including protocols of quantum teleportation, entanglement swapping and quantum cryptography. In its general form, our formula can be adopted to study quantum information protocols in the presence of experimental imperfections and asymmetric setups, e.g., deriving from the use of unbalanced beam splitters.