Asymmetric Gaussian steering: When Alice and Bob disagree

Asymmetric steering is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. This phenomenon is possible because of an asymmetry in the definition of the original Einstein-Podolsky-Rosen paradox and in this article we show theoretically that it may be demonstrated, at least in the case where Alice and Bob can only make Gaussian measurements, using the intracavity nonlinear coupler.

Figure 1

Schematic of the nonlinear coupler inside pumped coupled Fabry-Pérot cavities. The two ${\chi }^{(3)}$ materials acting as waveguides with nonlinear interaction strengths ${\chi }_i$ and evanescent coupling strength $J$ are shown, along with the external coherent fields with pump strengths ${\epsilon }_i$, cavity mirrors, and optical modes at frequency $\omega$.

Figure 2

The output spectral correlations ${\mathrm{EPR}}_{\mathit{ij}}$ for the parameters ${\gamma }_1=1$, ${\gamma }_2=36$, $J=5.0$, ${\Delta }_1=0.001J$, ${\Delta }_2=200{\Delta }_1$, ${\epsilon }_1={10}^3$, ${\epsilon }_2=80{\epsilon }_1$, ${\chi }_1={10}^{-8}$, and ${\chi }_2=10{\chi }_1$. For ${\mathrm{EPR}}_{12}$, $\theta =9^{°}$, while for ${\mathrm{EPR}}_{21}$, $\theta ={130}^{°}$, these angles giving the minimum values. We also show the Duan-Simon correlation for the same parameters, scaled so that a value of less than $1$ signifies bipartite entanglement. All quantities plotted in this and subsequent graphs are dimensionless.

Figure 3

(a) The minimum of ${\mathrm{EPR}}_{12}$ and (b) the minimum of ${\mathrm{EPR}}_{21}$ as functions of ${\epsilon }_2$ and ${\Delta }_2$, with other parameters the same as in Fig. 2. Considering (a) and (b) together, we observe regions of both symmetric and asymmetric steering.

Figure 4

(a) The minimum of ${\mathrm{EPR}}_{12}$ and (b) the minimum of ${\mathrm{EPR}}_{21}$ as functions of ${\gamma }_2$ and ${\chi }_2$, with other parameters the same as in Fig. 2. As in Fig. 3, we observe both symmetric and asymmetric steering in these plots.