Figure 3

The quantity to \(\pmb{\Delta\xi^{\prime2}}\) defined in Eq. ( 24 ), which describes the amount of correlations between Alice and Bob, plotted as a function of the transmission coefficients \(\pmb{\eta_{A,B}}\) modelling the losses in Alice’s and Bob’s channel. The case \(\Delta\xi^{\prime2}\geq1\) corresponds to a classically reachable performance. We see that the quality of the protocol depends on a compromise between squeezing, given by \(\Delta\xi^{2}\), and transmissivity coefficients, given by \(\eta_{A,B}\). (a) \(\Delta \hat{\xi}^{\prime2}\) plotted as a function of \(\eta_{B}\) for fixed \(\eta_{A}=0.70\) and for various values of \(\Delta\xi^{2}\), assuming \(\eta_{B}<\eta_{A}\) and noiseless EPR-JPAs. \(\Delta\xi^{2}\) determines the entanglement in the lossless case: the entanglement increases with decreasing \(\Delta\xi^{2}\). We see that the window of the allowed difference between the losses in Alice’s and Bob’s channel reduces for larger entanglement. (b) Here, \(\Delta\hat{\xi}^{\prime2}\) is plotted as function of \(\eta_{B}\) and \(\eta_{A}\) for fixed \(\Delta\xi^{2}=0.14\). From Eq. (24), we see that for a too large asymmetry between Alice’s and Bob’s channel, it is opportune to symmetrize them by attenuating one of the signals in order to increase the amount of correlations between the two parties. For instance, for \(\eta_{B}=0.3\) and \(0.8<\eta_{A}<1\), we find that \(\Delta\hat{\xi}^{\prime2}\) increases with increasing \(\eta_{A}\).