Fake violations of the quantum Bell-parameter bound

Shortcomings of experimental techniques are usually assumed to diminish nonclassical properties of quantum systems. Here it is demonstrated that this standard assumption is not true in general. It is theoretically shown that the inability to resolve different photon numbers with photodetection may pseudoincrease a measured Bell parameter. Under proper conditions one even pseudoviolates the quantum Cirel’son bound of the Bell parameter, and the corresponding density operator fails to be positive semidefinite. This paradox can be resolved by appropriate squash models.

Figure 1

A typical experimental setup for checking the violation of Bell inequalities. The source $S$ produces entangled photon pairs. The polarization analyzers $A$ and $B$ consist of half-wave plates (HWP), polarizing beam splitters (PBS), and two pairs of detectors: $D_{{\mathrm{T}}_{\mathrm{A}}}$ ($D_{{\mathrm{T}}_{\mathrm{B}}})$ for the transmitted signal and $D_{{\mathrm{R}}_{\mathrm{A}}}$ ($D_{{\mathrm{R}}_{\mathrm{B}}}$) for the reflected signal at the $A$ site ($B$ site).

Figure 2

Maximal value of the Bell parameter $\mathcal{B}$, obtained for on-off (solid lines) and photon-number-resolving (dashed lines) detectors, vs the parameter $\tanh \chi$ for the mean value of noise counts $N_{\mathrm{nc}}={10}^{-6}$ and the efficiencies (a) $\eta =0.9$, (b) $\eta =0.6$, and (c) $\eta =0.4$. The minimum at $\tanh \chi =0$ is caused by noise counts.

Figure 3

Maximal value of the Bell parameter $\mathcal{B}$, obtained for lossless on-off detectors without noise counts.

Figure 4

The minimum eigenvalue of the density operator $\mathrm{\rho ̂}$ [see Eq. (21)] vs the parameter $\tanh \chi$ for the mean value of noise counts $N_{\mathrm{nc}}={10}^{-6}$, the efficiency $\eta =0.6$, and the following sets of angles: (a) ${\theta }_{\mathrm{A}}^{(1)}={\theta }_{\mathrm{B}}^{(1)}=\pi /4$, ${\phi }_{\mathrm{A}}^{(1)}={\phi }_{\mathrm{B}}^{(1)}=0$, ${\theta }_{\mathrm{A}}^{(2)}={\theta }_{\mathrm{B}}^{(2)}=\pi /4$, ${\phi }_{\mathrm{A}}^{(2)}={\phi }_{\mathrm{B}}^{(2)}=\pi /2$, ${\theta }_{\mathrm{A}}^{(3)}={\theta }_{\mathrm{B}}^{(3)}=0$, ${\phi }_{\mathrm{A}}^{(3)}={\phi }_{\mathrm{B}}^{(3)}=0$; (b) ${\theta }_{\mathrm{A}}^{(1)}=\pi /8$, ${\phi }_{\mathrm{A}}^{(1)}=0$, ${\theta }_{\mathrm{A}}^{(2)}=9\pi /4$, ${\phi }_{\mathrm{A}}^{(2)}=\pi /2$, ${\theta }_{\mathrm{A}}^{(3)}=\pi$, ${\phi }_{\mathrm{A}}^{(3)}=0$, ${\theta }_{\mathrm{B}}^{(1)}=3\pi /15$, ${\phi }_{\mathrm{B}}^{(1)}=0$, ${\theta }_{\mathrm{B}}^{(2)}=-\pi /24$, ${\phi }_{\mathrm{B}}^{(2)}=\pi /2$, ${\theta }_{\mathrm{B}}^{(3)}=\pi$, ${\phi }_{\mathrm{B}}^{(3)}=0$.