Gaussian Intrinsic Entanglement

We introduce a cryptographically motivated quantifier of entanglement in bipartite Gaussian systems called Gaussian intrinsic entanglement (GIE). The GIE is defined as the optimized mutual information of a Gaussian distribution of outcomes of measurements on parts of a system, conditioned on the outcomes of a measurement on a purifying subsystem. We show that GIE vanishes only on separable states and exhibits monotonicity under Gaussian local trace-preserving operations and classical communication. In the two-mode case, we compute GIE for all pure states as well as for several important classes of symmetric and asymmetric mixed states. Surprisingly, in all of these cases, GIE is equal to Gaussian Rényi-2 entanglement. As GIE is operationally associated with the secret-key agreement protocol and can be computed for several important classes of states, it offers a compromise between computable and physically meaningful entanglement quantifiers.

Figure 1

(a) Construction of the purification of the output state ${\rho }_{A_{\mathrm{out}}{B}_{\mathrm{out}}}$ via teleportation. ${\mathrm{BM}}_j$: Bell measurement on subsystem $(j_{\mathrm{in}}{j}_1)$, $D_j$: displacement of subsystem $j_2$. (b) Decomposition of operation ${\mathcal{E}}_j$ and its integration into a measurement.