Gaussian entanglement of formation

We introduce a Gaussian version of the entanglement of formation adapted to bipartite Gaussian states by considering decompositions into pure Gaussian states only. We show that this quantity is an entanglement monotone under Gaussian operations and provide a simplified computation for states of arbitrary many modes. For the case of one mode per site the remaining variational problem can be solved analytically. If the considered state is in addition symmetric with respect to interchanging the two modes, we prove additivity of the considered entanglement measure. Moreover, in this case and considering only a single copy, our entanglement measure coincides with the true entanglement of formation.

Figure 1

For any two-mode Gaussian state with CM $\gamma =C_q\oplus C_p$ the Gaussian entanglement of formation is given by the entanglement of the least entangled pure state with CM ${\gamma }_p=X\oplus X^{-1}$ which is such that $C_p^{-1}⩽X⩽C_q$. Moreover, the optimal $X$ can be shown to lie on the rim of the intersection of the forward and backward cones of $C_p^{-1}$ and $C_q$, respectively.

Figure 2

Gaussian entanglement of formation (in units of ebits) for a TMSS with $r=1$ after transmission through a lossy optical fiber at temperature $\tau =0$. The plot shows $E_G\left(\rho \right)$ versus transmission length $l∕l_{\mathrm{A}}$ for the symmetric and asymmetric setting (dotted and solid line, respectively).

Figure 3

Gaussian entanglement of formation (in units of ebits) for a TMSS with $r=1$ after transmission through a lossy optical fiber at temperature $\tau =1$. The plot shows $E_G\left(\rho \right)$ versus transmission length $l∕l_{\mathrm{A}}$ for the symmetric and asymmetric setting (dotted and solid line, respectively).

Figure 4

Gaussian entanglement of formation (in units of ebits) for a TMSS with initial squeezing $r$ after symmetric transmission through a lossy optical fiber at zero temperature. $l∕l_{\mathrm{A}}$ is the transmission length in units of the absorption length.