Constraints on Gaussian error channels and measurements for quantum communication

Joint Gaussian measurements of two quantum systems can be used for communication between remote parties, as in teleportation or entanglement swapping protocols. Many types physical error sources throughout a protocol modeled by independent channels acting prior to measurement. In this work we study joint on modes $\mathsf{A}$ and $\mathsf{B}$ that take place after single-mode channels, example loss with parameters $l_\mathsf{A}$ $l_\mathsf{B}$ followed added noise $n_\mathsf{A}$ $n_\mathsf{B}$. We show that, any measurement, if $l_\mathsf{A} + l_\mathsf{B} n_\mathsf{A} n_\mathsf{B} \geq 1$ then the effective total measurement is separable unsuitable arbitrary input states. If inequality not satisfied there exists remains inseparable. extend results determine set pairs render all separable.