Violations of Bell’s inequality for Gaussian states with homodyne detection and nonlinear interactions

We show that homodyne measurements can be used to demonstrate violations of Bell’s inequality with Gaussian states, when the local rotations used for these types of tests are implemented using nonlinear unitary operations. We reveal that the local structure of the Gaussian state under scrutiny is crucial in the performance of the test. The effects of finite detection efficiency are thoroughly studied and shown to only mildly affect the revelation of Bell violations. We speculate that our approach may be extended to other applications such as entanglement distillation where local operations are necessary elements besides quantum entanglement.

Figure 1

(Color online) Schematic of a Bell-CHSH inequality test with Gaussian states and homodyne measurements. A quantum correlated two-mode Gaussian state is produced at a source and distributed to Alice and Bob who perform local effective rotations and homodyne measurements. The effective rotations are physically implemented by cascading Kerr-type nonlinearities and phase-space displacement operations, as shown in the inset. The Gaussian state produced by the source can be either a two-mode squeezed state or the state resulting from the superposition, at a 50:50 beam splitter, of vacuum and a single-mode squeezed state.

Figure 2

(Color online) Bell test for ideal rotations. The Bell-CHSH function is plotted against the squeezing parameter $r$ for three values of the detection efficiency $\eta$. We show the case corresponding to ideal homodyne detection (solid line), $\eta =0.5$ (dashed line), and $\eta =0.05$ (dotted line). The horizontal line shows the bound for local realistic theories.

Figure 3

(Color online) The numerically optimized Bell function is plotted against the squeezing parameter $r$ for the case of physical effective rotations and three values of detection efficiency. The horizontal line shows the bound for local realistic theories. The actual value of $d$ is irrelevant in this figure. The solid line with $\eta =1$ embodies the ideal-detector case with the other two cases of $\eta =0.8$ and $\eta =0.3$.

Figure 4

(Color online) We compare the behavior of the joint-probability functions ${\mid C_{id}({\theta }_A,{\theta }_B,x,y)\mid }^2$ and ${\mid {\mathcal{C}}_{ef}({\theta }_A,{\theta }_B,x,y)\mid }^2$ against the quadrature variables $x$ and $y$ for $r=4$. The angles ${\theta }_{A,B}$ are those maximizing the corresponding Bell-CHSH function. We thus have ${\theta }_A=0.061$ and ${\theta }_B=0.182$ (${\theta }_A=-0.009$ and ${\theta }_B=0.004$) for the leftmost (rightmost) plot. Moreover, $\int \int dxdy{\mid C_{ef}({\theta }_A,{\theta }_B,x,y)\mid }^2\simeq \int \int dxdy{\mid C_{id}({\theta }_A,{\theta }_B,x,y)\mid }^2$, regardless of the domain of integration.

Figure 5

(Color online) Bell-CHSH test for an input squeezed thermal state superimposed to vacuum at a 50:50 beam splitter. The Bell-CHSH function is plotted against $r$ and $V=2\overline{n}+1$, i.e., the thermal variance of the state. The horizontal plane shows the bound for local realistic theories.