Experimentally Robust Self-testing for Bipartite and Tripartite Entangled States

Self-testing is a method with which a classical user can certify the state and measurements of quantum systems in a device-independent way. In particular, self-testing of entangled states is of great importance in quantum information processing. An understandable example is that the maximal violation of the Clauser-Horne-Shimony-Holt inequality necessarily implies that the bipartite system shares a singlet. One essential question in self-testing is that, when one observes a nonmaximum violation, how far is the tested state from the target state (which maximally violates a certain Bell inequality)? The answer to this question describes the robustness of the used self-testing criterion, which is highly important in a practical sense. Recently, J. Kaniewski derived two analytic self-testing bounds for bipartite and tripartite systems. In this Letter, we experimentally investigate these two bounds with high-quality two-qubit and three-qubit entanglement sources. The results show that these bounds are valid for various entangled states that we prepared. Thereby, a proof-of-concept demonstration of robust self-testing is achieved, which improves on the previous results significantly.

Figure 1

Experimental setup. A periodically poled ${\text{KTiOPO}}_4$ (PPKTP) nonlinear crystal placed inside a phase-stable Sagnac interferometer (SI) is pumped by a single mode 405.4 nm laser to produce polarization-entangled photon pairs at 810.8 nm. Bandpass filters at 810 nm and long-pass filters are used to block the pump light. One photon will pass through a sufficiently unbalanced (to make the two light paths fully decoherent) Mach-Zehnder interferometer (MZI) constructed by two $50/50$ nonpolarizing beam splitters BS1 and BS2 when mixed states must be generated. By introducing replaceable modules on the long arm of this unbalanced MZI, different categories of states can be prepared. The other photon will pass through a balanced MZI constructed by beam displacers BD1 and BD2 when we need to generate tripartite states. One polarization beam splitter (PBS), a motorized half-wave plate (mHWP), and a motorized quarter-wave plate (mQWP) are used to perform the projection measurements on each qubit. BS, beam splitter; L, lense; Dia, diaphragm, Di, dichroic; IF, interfering filter; LBD, longtitudinal beam displacer; PCP, phase compensation plate.

Figure 2

Experimental test of operator inequality (1) with (a) different singlet states obtained by rotating HWP3 to the angles shown in the figure; (b) partially entangled states characterized by the value of $\theta$ shown in the figure; (c) mixed states obtained by inserting wave plates of different levels of retardation. With randomly sampled $\{a,b\}$, all data points are above the given bound. (d) Experimental self-testing of four special categories of entangled states, with the red and blue dashed lines representing the lower (self-testing) and upper bounds in inequality (2), respectively.

Figure 3

Experimental test of the self-testing bound for tripartite scenarios. (a) For several typical tripartite entangled states, the mean values of both the fidelity operator $K$ and the Mermin operator $W$ are measured, with random sampling on $(a,b,c)$. All the data points are above the bound (red line) given by operator inequality (3). (b) For mixtures of ${\mathrm{\Upsilon }}_{ABC}$ and ${\nu }_{ABC}$ with different weights, the maximum violations of Mermin inequality and corresponding extractabilities are measured. All the dots almost fall on the lower bound (red line), indicating the given bound is tight.

Figure 4

No-signaling tests for measuring the (a) CHSH inequality with singlet state and (b) Mermin inequality with the GHZ state. The mean value of each operator is invariant within one standard deviation, confirming the no-signaling characteristics of the used devices.