Testing quantum nonlocality by generalized quasiprobability functions

We derive a Bell inequality based on a generalized quasiprobability function, which is parametrized by one nonpositive real value. Two types of known Bell inequalities formulated in terms of the Wigner and the $Q$ functions are included as limiting cases. We investigate violations of our Bell inequalities for single-photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the $Q$ function allows the lowest detection efficiency for violations of local realism.

Figure 1

The optical setup for the BI test. Each local measurement is carried out after mixing the incoming field with a coherent state (denoted by $|\xi ⟩$ for Alice and $|\delta ⟩$ for Bob) in a beam splitter (BS) of high transmissivity $T$. The photon number detectors (PNDs) have efficiency ${\eta }_d$.

Figure 2

(Color online) Maximum Bell expectation value $\left|\mathcal{B}\right|=\left|⟨\hat{\mathcal{B}}⟩\right|$ for the single-photon entangled state. Only the range of parameters $s$ and detector efficiencies $\eta$ with $\left|\mathcal{B}\right|>2$ are shown.

Figure 3

(Color online) Demonstration of quantum nonlocality for TMSSs. (a) Maximum Bell values are shown for different squeezing $r$ in the range of $s$ and $\eta$ where the BI is violated. (b) Violation of the BI as a function of the squeezing $r$ for different $s$ and $\eta =1$ (solid line), $\eta =0.95$ (dashed line), and $\eta =0.9$ (dotted line).