Testing nonlocal realism with entangled coherent states

We investigate the violation of nonlocal realism using entangled coherent states (ECSs) under nonlinear operations and homodyne measurements. We address recently proposed Leggett-type inequalities, including a class of optimized incompatibility inequalities proposed by Branciard et al. [Nature Phys. 4, 681 (2008)], and thoroughly assess the effects of detection inefficiency.

Figure 1

Scheme of the experiment for testing nonlocal realism with an ECS. The source generates an ECS of the form considered in the body of the article. Alice and Bob perform local rotations through the sequence of unitary operations in Eq. (3). The locally rotated optical states are then mixed with a strong local oscillator (LO), and homodyne detectors (HDs) are used for final measurements. The leftmost and rightmost spheres show the directions of the vectors identifying the measurement settings at Alice’s and Bob’s sites, respectively.

Figure 2

(a) Violation of nonlocal realism by the Leggett function $L$ in Eq. (10) for $\alpha =10$ (dashed curve) and $\alpha =60$ (solid curve). The upper bound for the Leggett model is also plotted against the angle $\phi$. The vertical dashed line indicates the value $\phi \simeq 0.2507$ rad at which the Leggett-type inequality in [8] is maximally violated, regardless of $\alpha$. (b) Violation of Leggett’s model by $L$ against the amplitude of the coherent-state component in the ECS considered in the body of the article. In this plot, we have assumed $\phi =0.25$. The upped bound for Leggett model (straight line) is surpassed for $\alpha ⩾7.5$.

Figure 3

Violation of local and nonlocal realism against the amplitude $\alpha$ of the ECS components. We show the Bell function $B$ (together with the local realistic bound $2$) and the function $\mathcal{L}$, which violates the Leggett-type inequality when positive. The shaded region corresponds to values of $\alpha$ where local realistic theories should be abandoned while Leggett’s inequality is still satisfied.

Figure 4

Measurement settings at Alice and Bob’s site for the test of nonlocal realism as proposed by Branciad et al. in Ref. [8]. The pairs of vectors $({\mathbf{b}}_i,{\mathbf{b}}_i^{\prime })$ with $i=1,2,3$ lie on orthogonal planes, and the vectors form angles equal to $\phi$. The pair with $i=1,2,3$ lies on the plane with $z,x,y=0$, respectively.

Figure 5

Violation of a Leggett-type inequality by the function $L_S$ in Eq. (11) for $\alpha =60$. The upper bound for Leggett model is also plotted against the angle $\phi$. The leftmost vertical dashed line indicates the value $\phi \simeq 0.65$ rad at which the Leggett-type inequality in [8] is maximally violated. The rightmost dashed line, corresponding to $\phi \simeq 1.28$ rad, sets the upper bound for $\phi \in [0,\pi /2]$ for the Leggett-type inequality.

Figure 6

(a) Leggett function with inefficient detection $L^d$ plotted against the coherent-state amplitude $\alpha$ and the detection inefficiency $\eta$. The horizontal plane is the nonlocal realistic bound for the inequality in Refs. [6, 7] at $\phi \simeq 0.25$. (b) $L^d$ versus $\alpha$ for decreasing values of $\eta$, which goes from $1$ to $0.1$ (in steps of $0.1$) from the leftmost to the rightmost curve. The straight line indicates the nonlocal realistic bound.