Experimental Demonstration of the Einstein-Podolsky-Rosen Steering Game Based on the All-Versus-Nothing Proof

Einstein-Podolsky-Rosen (EPR) steering, a generalization of the original concept of “steering” proposed by Schrödinger, describes the ability of one system to nonlocally affect another system’s states through local measurements. Some experimental efforts to test EPR steering in terms of inequalities have been made, which usually require many measurement settings. Analogy to the “all-versus-nothing” (AVN) proof of Bell’s theorem without inequalities, testing steerability without inequalities would be more strong and require less resources. Moreover, the practical meaning of steering implies that it should also be possible to store the state information on the side to be steered, a result that has not yet been experimentally demonstrated. Using a recent AVN criterion for two-qubit entangled states, we experimentally implement a practical steering game using quantum memory. Furthermore, we develop a theoretical method to deal with the noise and finite measurement statistics within the AVN framework and apply it to analyze the experimental data. Our results clearly show the facilitation of the AVN criterion for testing steerability and provide a particularly strong perspective for understanding EPR steering.

Figure 1

Illustration of the EPR steering game. (1) Alice measures her own qubit along the direction $\overrightarrow{n}$ and obtains the outcomes of $+1$ and $-1$. (2) Through a classical channel, Alice tells Bob her measurement outcome and the corresponding output state that Bob should obtain. (3) Bob verifies the normalized conditional states. (4) Alice and Bob implement a joint measurement. They determine the value of $⟨\mathcal{W}{⟩}_{\max }$ and compare it with the upper bound predicted by the LHS model ($C_{\mathrm{LHS}}$).

Figure 2

Experimental setup. The entangled photon pairs are produced via SPDC. An unbalanced Mach-Zehnder (UMZ) interferometer (a) is employed to prepare the states ${\rho }_1$. The state ${\rho }_2$ is prepared by inserting UMZ (b) consisting of two beam displacers (BD) and a quartz plate (QP) in the long arm of UMZ (a).

Figure 3

Experimental results for ${\rho }_1$. (a)–(d) show the detected probabilities of the NCS on Bob’s side. The hollow points represent the states are not steerable in the case of two measurement settings based on the values of ${\mathrm{\Delta }}^{\prime }$ which are shown in (f), while the solid ones mean the states are steerable. The blue squares and black circles represent the values of $P_{+}$ and $P_{+}^{\prime }$, respectively. The green up triangles and red stars represent the values of $P_{-}$ and $P_{-}^{\prime }$, respectively. (e) The values of $⟨\mathcal{W}⟩$ as a function of ${\theta }_B$. The black circles, green up triangles, magenta diamonds, and blue stars represent cases with input parameters of $\theta =\pi /4$ and $\eta =1$, $\theta =\pi /3$ and $\eta =0.2$, $\theta =\pi /6$ and $\eta =0$, and $\theta =\pi /3$ and $\eta =1$, respectively. The black, red, green, magenta, and blue lines represent the corresponding theoretical predictions. (f) The results for ${\mathrm{\Delta }}^{\prime }$. The red triangles, blue squares, and green circles represent the cases with initial parameters of $\theta =\pi /6$, $\theta =\pi /3$, and $\theta =\pi /4$, respectively. The inset in (f) shows the value of ${\mathrm{\Delta }}^{\prime }$ as a function of $\theta$. The red circles represent the cases with initial parameters of $\eta =1$. The error bars correspond to the counting statistics.

Figure 4

Experimental results for ${\rho }_2$. (a)–(d) show the detected probabilities of the NCS on Bob’s side. The hollow points represent the states are not steerable in the case of two measurement settings based on the values of ${\mathrm{\Delta }}^{\prime }$ which are shown in (f), while the solid ones mean the states are steerable. The blue squares and black circles represent the values of $P_{+}$ and $P_{+}^{\prime }$, respectively. The green up triangles and red stars represent the values of $P_{-}$ and $P_{-}^{\prime }$, respectively. (e) The values of $⟨\mathcal{W}⟩$ as a function of ${\theta }_B$. The black circles, red up triangles, green diamonds, and magenta stars represent the cases with input parameters of $\theta =\pi /4$ and $\eta =1$, $\theta =\pi /2$ and $\eta =1$, $\theta =\pi /6$ and $\eta =0.8$, and $\theta =\pi /3$ and $\eta =0.7$, respectively. The black, red, green, and magenta lines represent the corresponding theoretical predictions. (f) The results for ${\mathrm{\Delta }}^{\prime }$. The red triangles, blue squares, and green circles represent the cases with initial parameters of $\theta =\pi /6$, $\theta =\pi /3$, and $\theta =\pi /4$, respectively. The inset in (f) shows the value of ${\mathrm{\Delta }}^{\prime }$ as a function of $\theta$. The red circles represent the case with an initial parameter of $\eta =1$. The error bars correspond to the counting statistics.