Continuous-variable Clauser-Horne Bell-type inequality: A tool to unearth the nonlocality of continuous-variable quantum-optical systems

We consider a continuous-variable Clauser-Horne Bell-type inequality to study nonlocality in four-mode continuous-variable systems, which goes beyond two-photon states and can be applied to mixed as well as to states with fluctuating photon number. We apply the inequality to a wide variety of states such as pure and mixed Gaussian states (including squeezed thermal states) and non-Gaussian states. We consider beam splitters as a model for leakage and show that the inequality is able to detect nonlocality of noisy Gaussian states as well. Finally, we investigate nonlocality in pair-coherent states and entangled coherent states, which are prominent examples of nonclassical, non-Gaussian states.

Figure 1

Setup to study Bell inequality violation for states of a four-mode radiation field.

Figure 2

Schematic to generate a four-mode entangled state. Here $S\left(u\right)$ and $S\left(v\right)$ represent squeezing transformations. Further, $U_1$, $U_2$, $V_1$, and $V_2$ represent transformations that can be generated by combinations of quarter- and half-wave plates and phase shifters, while $D$ represents transformations that can be generated using beam splitters and quarter- and half-wave plates. The first part of the circuit generates nonclassicality by squeezing the individual modes and the passive operations comprising beam splitter, phase shifters, and quarter- and half-wave plates convert the nonclassicality into entanglement. $\text{Classical}\stackrel{\text{Squeezing}}{\to }\text{Nonclassical}\stackrel{\text{Passive}\text{operations}}{\to }\text{Entangled}$.

Figure 3

Average of Bell operator as a function of squeezing parameter $u$ for the four-mode pure squeezed vacuum state. Different angles are fixed as ${\theta }_1=1.32$, ${\theta }_2=0.93$, ${\theta }_1^{\prime }=3.66$, ${\theta }_2^{\prime }=3.32$. Thick solid line represents the case $v=-u$ corresponding to the state ${|\text{TMSV}\rangle }_{13}{|\text{TMSV}\rangle }_{24}$ showing violation of the CV-CH Bell-type inequality. Dashed line corresponds to $v=0$, which also violates the inequality although to a less extent.

Figure 4

Average of the Bell operator as a function of squeezing parameter $u$ for the four-mode squeezed thermal state for the case $v=-u$. Different angles are fixed as ${\theta }_1=1.32$, ${\theta }_2=0.93$, ${\theta }_1^{\prime }=3.66$, ${\theta }_2^{\prime }=3.32$. Thick solid, dashed, and thin solid graphs depict $\kappa =1$, $\kappa =0.8$, and $\kappa =0.7$, respectively. Results show that an increase in temperature results in loss of nonlocal correlations.

Figure 5

Modeling leakage with beam splitters. The input state of the system is ${|\text{TMSV}\rangle }_{13}{|\text{TMSV}\rangle }_{24}$, which maximally violates the CV-CH Bell-type inequality. Each mode of the pure input state is mixed with vacuum using two beam splitters of transmittance $T$. Subsequently, mode corresponding to vacuum is discarded, and thus the output is a mixed state.

Figure 6

Average of Bell operator as a function of squeezing parameter $u$ for a four-mode pure squeezed vacuum state ${|\text{TMSV}\rangle }_{13}{|\text{TMSV}\rangle }_{24}$ in the presence of leakage. Different angles are fixed as ${\theta }_1=1.32$, ${\theta }_2=0.93$, ${\theta }_1^{\prime }=3.66$, ${\theta }_2^{\prime }=3.32$. Thick solid, dashed, and thin solid lines correspond to transmittance, T = 1, 0.8, and 0.6, respectively. Results indicate that the nonlocal character of state remains preserved under leakage.

Figure 7

Average of Bell operator as a function of parameter $\text{Re}\left(\zeta \right)$ for the pair coherent state with $q=0$. The angles are optimized to obtain maximum violation. The result shows that the pair coherent state violates CV-CH Bell-type inequality.

Figure 8

Average of Bell operator as a function of $\text{Re}\left(\alpha \right)$ for the entangled coherent state. Different angles are fixed as ${\theta }_1=2.67$, ${\theta }_2=5.59$, ${\theta }_1^{\prime }=1.88$, ${\theta }_2^{\prime }=3.24$. The results clearly indicates that the entangled coherent state violates CV-CH Bell-type inequality.

Figure 9

Average of the Bell operator as a function of the squeezing parameter $u$ of the TMSV state. Any quantum state violating the inequality $\left|\mathcal{CHSH}\right|\le 2$ or $-1\le \mathcal{CH}\le 0$ is said to be nonlocal. The results show that all the four Bell-type inequalities can detect nonlocality in the TMSV state. (a) Bell-CHSH inequality based on displaced parity operator measurements given by Banaszek et al. [34]. (b) Bell-CHSH inequality based on pseudospin measurements given by Chen et al. [35]. (c) CH inequality based on displaced on-off measurements given by Banaszek et al. [34]. (d) CH inequality based on on-off measurements given by Arvind et al. [33].

Figure 10

Average of the Bell operator as a function of displacement “d” before on-off measurements. Any quantum state violating the inequality $-1\le \mathcal{CH}\le 0$ is said to be nonlocal. (a) For the singlet state. (b) For the entangled coherent state.