Classifying Directional Gaussian Entanglement, Einstein-Podolsky-Rosen Steering, and Discord

Using Venn diagrams, we classify the different types of two-mode Gaussian continuous variable quantum correlation including directional entanglement and Einstein-Podolsky-Rosen (EPR) steering. We establish unified signatures for one- and two-way quantum steering, entanglement, and discord beyond entanglement in terms of an EPR-type variance. By focusing on Gaussian states, we link an optimized condition for entanglement based on an EPR variance to the Simon-Peres condition. This allows us to quantify the asymmetry of the Gaussian entanglement, and to relate the asymmetry to a directional quantum teleportation protocol where Alice and Bob possess asymmetrically noisy channels. Our analysis enables a determination of the type and direction of quantum correlation in a way that is easily measured in experiment. We also find that for symmetric states, when discord exceeds a certain threshold, the states are necessarily steerable.

Figure 1

The Venn diagram relations classifying the different types of quantum correlation for two-mode, two-party Gaussian states. The larger blue circle II contains states satisfying the Tan-Duan criterion for entanglement ${\mathrm{\Delta }}_{\mathrm{ent}}<1$. The inner blue circle I contains states with the symmetric two-way EPR steering correlation given by ${\mathrm{\Delta }}_{\mathrm{ent}}<0.5$. The set of all entangled states quantified by the Simon-PPT criterion ${\mathrm{Ent}}_{\mathrm{PPT}}<0$ (equivalent to $\mathrm{Ent}<1$) are contained in the larger green ellipse VI. The smaller orange-pink IV and yellow V ellipses enclose states that display the directional steering, given by the EPR-Reid paradox condition $E_{A|B}<1$ and $E_{B|A}<1$, respectively. Their intersection (colored dark yellow) is the set of two-way steerable states, which is a strict superset of the states in I. All two-way steerable states are a subset of the entangled states quantified by the Tan-Duan condition ${\mathrm{\Delta }}_{\mathrm{ent}}<1$. One-way steering states are a strict subset of the PPT entangled states, and are strictly not contained in the Tan-Duan circle ${\mathrm{\Delta }}_{\mathrm{ent}}<1$. The symmetric states with $g_{\mathrm{sym}}^{A|B}=1$ ($n=m$) are depicted at the very center of the diagram (dashed line). Those with $g_{\mathrm{sym}}^{A|B}>1$ ($n>m$) are to the right of the centre line; those with $g_{\mathrm{sym}}^{A|B}<1$ ($n<m$) to the left. The outer ellipse III contains the set of Gaussian states with nonzero quantum $A$ and $B$ discord.

Figure 2

Contour plots show the effect of asymmetric noises $n_A$ and $n_B$ on quantum correlation, for the two-mode STS with $r=0.6$: (a) entanglement measured by Ent, (b) discord measured by $D_{A|B}$, and (c) the steering parameter $E_{A|B}$. In (c), the states can be used as quantum resources with EPR steering (below red curve), entanglement (below green curve), or discord beyond entanglement (above green curve) as explained in text. (d) shows the different regions defined in Fig. 1 certified by criteria of steering, entanglement, discord, and unphysical CMs (light gray area $UP$) versus $n$ and $m$.

Figure 3

The allowed Gaussian steerability (shaded region) versus discord for the symmetric Gaussian case restricting to $c_1=-c_2=c$ and $n=m$.