order for the map to be a completely positive, trace-preserving map (i.e., a legitimate quantum channel), the following inequality should hold [@S17]$$y\geq\left\vert 1-\tau\right\vert .$$ The main contribution of [@LMGA17] is that every single-mode phase-insensitive Gaussian channel in the above class, besides the pure-loss channel, can be simulated by the action of a continuous-variable teleportation protocol on a finite-energy, two-mode resource state with the same amount of entanglement as the Choi state of the channel. An additional contribution of [@LMGA17] is a converse bound: it is not possible to use a resource state with logarithmic negativity smaller than that of the Choi state, in order to simulate the channel. This follows directly from the facts that the teleportation simulation protocol should simulate the channel, teleportation is an LOCC, and logarithmic negativity is an entanglement monotone (it is non-increasing with respect to an LOCC). This converse bound holds, by the same argument, for all measures of entanglement (such as relative entropy of entanglement).

In more detail, the teleportation simulation of [@LMGA17] begins with the sender and receiver of the channel sharing a two-mode Gaussian state in the standard form in . The sender mixes the input of the channel and her share 