Realistic limits on the nonlocality of an $N$-partite single-photon superposition

A recent paper [L. Heaney, A. Cabello, M. F. Santos, and V. Vedral, New J. Phys. 13, 053054 (2011)] revealed that a single quantum symmetrically delocalized over $N$ modes, namely a $W$ state, effectively allows for all-versus-nothing proofs of nonlocality in the limit of large $N$. Ideally, this finding opens up the possibility of using the robustness of the $W$ states while realizing the nonlocal behavior previously thought to be exclusive to the more complex class of Greenberger-Horne-Zeilinger states. We show that in practice, however, the slightest decoherence or inefficiency of the Bell measurements on $W$ states will degrade any violation margin gained by scaling to higher $N$. The nonstatistical demonstration of nonlocality is thus proved to be impossible in any realistic experiment.

Figure 1

Experimental setup for the tripartite example ($N=3$). (a) Conceptual setup for the preparation of the $W$ state. A single photon is fed into two consecutive beam splitters of reflectivities $\frac{1}{3}$ and $\frac{1}{2}$, respectively. The generation of the input single photon can be achieved by heralding one photon from the pair emitted via spontaneous parametric down conversion. Any inefficiency of the heralding detector will not affect the purity of the produced state, but will simply reduce its generation rate. (b) Measurement scheme. Each of the three modes that emerge are randomly measured in either one of two ways: by a projection on $\widehat{Z}$ or by a projection on $\widehat{X}$. Fictitious beam splitters, drawn here in gray, are merely used as mathematical models for inefficient detection. If one works with optimal detectors, then the effectively measured state is given by Eq. (3). (c) Remote preparation of the $W$ state. Three weak squeezers (labeled ${\chi }^2$) produce a state $p_0|00\rangle +p_1|11\rangle$ (with $p_0\gg p_1$) where one mode from each is sent “backward” to the same beam splitter arrangement as in (a) for entanglement with the other two. Upon detection of a single photon, the remote modes collapse to a $W$ state.

Figure 2

Scaling of the Bell factor with the number $N$ of delocalized modes. Four scenarios are considered regarding the detection inefficiencies: ideal detection (squares), 10$\%$ detection inefficiency in $\widehat{Z}$ (circles), 10$\%$ detection inefficiency in $\widehat{X}$ (upward triangles), and 10$\%$ detection inefficiency in both $\widehat{Z}$ and $\widehat{X}$ (downward triangles). It is worth noting that inefficiencies in $\widehat{Z}$ are more detrimental to the Bell factor than inefficiencies in $\widehat{X}$.

Figure 3

Minimum quantum efficiency for either $\widehat{Z}$ or $\widehat{X}$ as a function of the number of modes. These plots are obtained by solving $\Omega =0$ in Eq. (13) for ${\eta }_z^2$ (squares) and ${\eta }_x^2$ (circles) while maintaining ${\eta }_x^2=1$ and ${\eta }_z^2=1$, respectively. The top curve traces the minimum quantum efficiency required of photon detectors if one is to use a Hadamard rotation to perform $\widehat{X}$ measurements from a $\widehat{Z}$ basis (triangles).

Figure 4

Quadrature probability distribution of the qubit $\cos \left(\theta \right)|0\rangle +\sin \left(\theta \right)|1\rangle$ for $\theta =\frac{\pi }{4}$ and $\frac{7\pi }{4}$.

Figure 5

Scaling of the Bell factor with the number $N$ of delocalized modes in the case of hybrid detection involving an APD for $\widehat{Z}$ and homodyne thresholding for $\widehat{X}$. Four scenarios are considered regarding the detection inefficiencies: ideal detection (squares), 90$\%$ quantum efficiency for the APD (circles), 90$\%$ detection efficiency for the homodyne detector (upward triangles), and 90$\%$ detection efficiency for both detectors (downward triangles).

Figure 6

Minimum quantum efficiency for either the APD or the homodyne detector (HD) as a function of the number of modes. These plots are obtained by solving ${\Omega }_{\mathrm{hybrid}}=0$ in Eq. (18) for ${\eta }_{\mathrm{APD}}^2$ (squares) and ${\eta }_{\mathrm{HD}}^2$ (circles) while maintaining ${\eta }_{\mathrm{HD}}^2=1$ and ${\eta }_{\mathrm{APD}}^2=1$, respectively.