Using macroscopic entanglement to close the detection loophole in Bell-inequality tests

We consider a Bell-like inequality performed using various instances of multiphoton entangled states to demonstrate that losses occurring after the unitary transformations used in the nonlocality test can be counteracted by enhancing the size of such entangled states. In turn, this feature can be used to overcome detection inefficiencies affecting the test itself: a slight increase in the size of such states, pushing them towards a more macroscopic form of entanglement, significantly improves the state robustness against detection inefficiency, thus easing the closing of the detection loophole. Differently, losses before the unitary transformations cause decoherence effects that cannot be compensated using macroscopic entanglement.

Figure 1

(a) Bell inequality test using entangled polarization state and on/off photodetectors under photon loss. States $|n_H\rangle$ and $|n_V\rangle$ are discriminated by Alice using a polarization beam splitter (PBS) and two photodetectors $A$ and $B$. A local unitary operation, $U$, is required when performing a Bell inequality test. Photon loss is modeled by a beam splitter (BS) with transmittivity $\eta$. BS1 and BS2 are employed to consider losses both before and after the unitary operation, respectively. Bob undergoes the same procedure with his chosen unitary operation. (b) Analogous setup for entangled coherent state, $|\mathrm{ECS}\rangle$, and homodyne detection. Two component states, $|\alpha \rangle$ and $|-\alpha \rangle$, of an entangled coherent state are distinguished by homodyne detection.

Figure 2

The optimal Bell function of the entangled polarization state against $\eta$ for two cases depicted in Fig. 1 (a). (a) The case of photon losses after the unitary operation that is equivalent to inefficient detection. As photon number $n$ increases, nonlocality shown by Bell-inequality violations gets more robust against photon losses. The detection-inefficiency threshold decreases to 35.6$\%$ for $n=4$. (b) The case of photon losses before the unitary operation.

Figure 3

The optimized Bell parameter for an ECS. Similarly to polarization-entangled states, by increasing the macroscopic character of the state (given in this case by the amplitude $\alpha$), the Bell parameter becomes more robust for photon losses. For $\alpha =1$, the detection-inefficiency threshold is about $50\%$.

Figure 4

(a) The optimized Bell function against the dimensionless decoherence parameter $\gamma t$ and $d$ for $V=10$. The floor of the figure shows the limit for local realistic theories. (b) The optimized Bell function against $\gamma t$ for $V=1.001$, $d=5$ (solid curve), $V=10$, $d=5$ (dashed curve), and $V=10$, $d=10$ (dotted curve).

Figure 5

(a) The optimal Bell function for three values of $n$ in an entangled polarization state plotted against $\eta$ when 5$\%$ losses occur before the unitary operation. (b) The analogous quantity calculated for an ECS having $\alpha =1,1.5,$ and 2 and subjected to $15\%$ losses.