Witnessing entanglement in phase space using inefficient detectors

We propose a scheme for witnessing entanglement in phase space by significantly inefficient detectors. The implementation of this scheme does not require any additional process for correcting errors, in contrast to previous proposals. Moreover, it allows us to detect entanglement without full a priori knowledge of the detection efficiency. It is shown that entanglement in single-photon entangled and two-mode squeezed states can be witnessed with detection efficiency as low as 40%. Our approach enhances the possibility of witnessing entanglement in various physical systems using current detection technologies.

Figure 1

Maximum expectation value of the EW operator in Eq. (5) for an input of (a) a single-photon entangled state and (b) a two-mode squeezed state with $r=0.4$ (black) and $r=0.8$ (gray). Entanglement exists if the expectation value exceeds the dashed line ${\mathcal{W}}_{\max }^{\mathrm{sep}}=2$. Note that the shaded region which exceeds ${\mathcal{B}}_{\max }^{\mathrm{LR}}=2(1-2/\eta ){}^2$ (for $1/2<\eta ⩽1$) is the criterion of nonlocality. (c) Witnessing entanglement with varying squeezing rate $r$ of a two-mode squeezed states for detector efficiencies $\eta =1$ (solid line), $\eta =0.99$ (dashed line), $\eta =0.7$ (dot-dashed line), and $\eta =0.5$ (dotted line).

Figure 2

Witnessing entanglement with a real efficiency $\eta =0.55$ when varying the estimated efficiency $\varepsilon$ for an input of (a) a single-photon entangled state and (b) a two-mode squeezed state with $r=0.4$ (black) and $r=0.8$ (gray).