Covariance matrices under Bell-like detections

TL;DR

This paper presents a simple, versatile formula for how covariance matrices of bosonic modes transform under Bell-like detections, including realistic imperfections like detector inefficiency, aiding quantum information protocol analysis.

Contribution

The authors derive a general, easy-to-use formula for covariance matrix transformations under Bell-like detections, accommodating various beam splitter parameters and detector efficiencies.

Findings

Formula applies to standard Bell and heterodyne detections

Includes effects of non-unit quantum efficiency

Useful for analyzing realistic quantum protocols

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.