Constraints on continuous variable entanglement swapping in the presence of photon loss

In many procedures for entanglement swapping in the continuous variable domain the entangling measurement is quadrature measurement of linear mixtures of the two modes sent by the two parties that wish to swap entanglement. In the presence of sufficient photon loss, which we model as a fictitious beamsplitter with the environment on each of the two modes, we show that such a measurement cannot be used to swap entanglement. We show analytically that if the modes sent by the two parties suffer losses that average between them to 50% or more, then any passive linear circuit followed by quadrature measurement has a separable POVM on those modes and therefore cannot be used to perform entanglement swapping. We will also discuss the case where the constraints on the measurement are relaxed to include arbitrary Gaussian resources.