Experimental detection of quantum steerability based on the critical radius in an all-optical system

Quantum steering is the ability that one system affects another one without delay by performing local measurements. During the past few years, the research on quantum steering has been verified in a number of experiments by the choices of measurements. But this research cannot detect all steerable states. Recently, the definition of critical radius was proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)]. The critical radius can detect all steerable states for two-qubit states. Therefore the critical radius can be regarded as the best method for detecting the steerability of two-qubit states. However, relevant experimental research is still lacking. In this work, for comparison, we experimentally investigate three steering criteria based on the critical radius, the geometric Bell-like inequality, and the three-setting measurements steering inequality and find that the critical radius can detect the most steerable states. Furthermore, we demonstrate that the critical radius satisfies the scaling property. The results show that we only need to calculate the critical radii of some quantum states to obtain the critical radii of more scaling quantum states. Thus this property can provide the convenience for detecting the steerability of many quantum states.

Figure 1

Experimental setup. (a) Entanglement source module. The maximum entangled state $|{\varphi }_1\rangle =\frac{1}{\sqrt{2}}\left(|HH\rangle +|VV\rangle \right)$ can be generated. (b) Phase reversal mixing state module. We can obtain the mixed state ${\rho }_{\text{BDS}}\left(1,q\right)=q|{\varphi }_1\rangle \langle {\varphi }_1|+\left(1-q\right)|{\varphi }_2\rangle \langle {\varphi }_2|$, where $|{\varphi }_2\rangle =\frac{1}{\sqrt{2}}\left(|HV\rangle +|VH\rangle \right)$ is the maximum entangled state formed by the phase inversion. (c) Depolarization mixing state module. We can finally prepare the mixed state ${\rho }_{\text{BDS}}\left(p,q\right)=p{\rho }_{\text{BDS}}\left(1,q\right)+\left(1-p\right)\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2}=q{\rho }_1\left(p\right)+\left(1-q\right){\rho }_2\left(p\right)$, where ${\rho }_1\left(p\right)=p|{\varphi }_1\rangle \langle {\varphi }_1|+\left(1-p\right)\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2}$ and ${\rho }_2\left(p\right)=p|{\varphi }_2\rangle \langle {\varphi }_2|+\left(1-p\right)\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2}$ are two typical Werner states. (d) State measurement module. We can obtain the measurement probability under any measurement basis. Key components: BBO, barium borate crystal; MIR, mirror; HWP, half-wave plate; QWP, quarter-wave plate; BS, beam splitter; PBS, polarizing beam splitter; ATT, attenuator; IF, interference filter; SPD, single photon detector.

Figure 2

Experimental results of steering criteria based on the geometric Bell-like inequality, the three-setting measurements steering inequality, and the critical radius with the different parameter $p$: (a) $p=0.5$, (b) $p=0.6$, (c) $p=0.7$, (d) $p=0.8$, (e) $p=0.9$, and (f) $p=1$. Red dotted lines, green dashed lines, and blue solid lines in every subfigure stand for the theoretical results of $F_Q, F_M$, and $F_R$ in Eq. (12). The value of the black dashed lines is the constant 1, which represents the right-hand side of the above steering criteria. In addition, red triangles, green prisms, and blue dots, which are obtained by our experimental data, stand for $F_Q, F_M$, and $F_R$, respectively.

Figure 3

The distribution of steering and unsteering for the states ${\rho }_{\text{BDS}}\left(p,q\right)$. The blue line denotes Eq. (13), which corresponds to the theoretical boundary between steering and unsteering. The steering areas are above the blue line, and the unsteering areas are below the blue line. In addition, the green rhombus represents the prepared states with $F_R>1$ and the red dots represent the prepared states with $F_R⩽1$.

Figure 4

The verification of the scaling property. Here, $y$ and $x$ represent $F_R\left[{\rho }_{\text{BDS}}\left(p,q\right)\right]$ and $F_R\left[{\rho }_{\text{BDS}}\left(1,q\right)\right]$, respectively. The red line, purple line, green line, cyan line, and blue line are the function $y=px$ in the cases $p=0.5,0.6,0.7,0.8,0.9$, respectively, which stand for the theoretical results. In addition, we use red dots, purple squares, green prisms, cyan pentagrams, and blue triangles to represent the experimental results of cases $p=0.5,0.6,0.7,0.8,0.9$, respectively.