Adjusting inequalities for detection-loophole-free steering experiments

We study the problem of certifying quantum steering in a detection-loophole-free manner in experimental situations that require postselection. We present a method to find the modified local-hidden-state bound of steering inequalities in such a postselected scenario. We then present a construction of linear steering inequalities in arbitrary finite dimension and show that they certify steering in a loophole-free manner as long as the detection efficiencies are above the known bound below which steering can never be demonstrated. We also show how our method extends to the scenarios of multipartite steering and Bell nonlocality, in the general case where there can be correlations between the losses of the different parties. In both cases we present examples to demonstrate the techniques developed.

Figure 1

Graphs of ${\beta }_{\mathrm{ps}}^{l}hs\left(\mathbf{\eta }\right)$ as a function of $\eta$, for the case of isotropic and uncorrelated loss, ${\eta }_x=\eta , {\eta }_y=\eta , {\eta }_{xy}={\eta }^2$ (solid green curve). An observation of $\beta >{\beta }_{\mathrm{ps}}^{l}hs\left(\mathbf{\eta }\right)$ when the losses are $\mathbf{\eta }$ certifies that multipartite steering has been witnessed in a detection-loophole-free manner. The steering functionals used are given in Eqs. (39) and (40), respectively. The red lines show the value obtained by the assemblage arising from the GHZ and $W$ state, respectively, when the uncharacterized parties perform the three Pauli measurements. The dotted black line shows the critical detection efficiency, below which steering is not demonstrated in a loophole-free manner.

Figure 2

Graphs of ${\beta }_{\mathrm{ps}}^{l}hv\left(\mathbf{\eta }\right)$ as a function of $\eta$, for the case of (a) isotropic and uncorrelated loss ${\eta }_x=\eta , {\eta }_y=\eta , {\eta }_{xy}={\eta }^2$ and (b) one-sided isotropic loss ${\eta }_x=1, {\eta }_y=\eta , {\eta }_{x}y=\eta$. An observation of $\beta >{\beta }_{\mathrm{ps}}^{l}hv\left(\mathbf{\eta }\right)$ when the losses are $\mathbf{\eta }$ certifies that nonlocality has been witnessed in a detection-loophole-free manner. The Bell functional used is $I_{\alpha }=\alpha \langle A_1{B}_1+A_2{B}_1\rangle +\langle A_1{B}_2-A_2{B}_2\rangle$, for different values of $\alpha$, in a bipartite Bell scenario where Alice and Bob perform two dichotomic measurements. In case (a), the Eberhard bound $\beta \ge 2/3$ is recovered. In case (b), nonlocality can be witnessed in a loophole-free manner whenever $\beta >1/2$.