Experimental Measurement-Device-Independent Quantum Steering and Randomness Generation Beyond Qubits

In a measurement-device-independent or quantum-refereed protocol, a referee can verify whether two parties share entanglement or Einstein-Podolsky-Rosen (EPR) steering without the need to trust either of the parties or their devices. The need for trusting a party is substituted by a quantum channel between the referee and that party, through which the referee encodes the measurements to be performed on that party’s subsystem in a set of nonorthogonal quantum states. In this Letter, an EPR-steering inequality is adapted as a quantum-refereed EPR-steering witness, and the trust-free experimental verification of higher dimensional quantum steering is reported via preparing a class of entangled photonic qutrits. Further, with two measurement settings, we extract $1.106\pm 0.023$ bits of private randomness per every photon pair from our observed data, which surpasses the one-bit limit for projective measurements performed on qubit systems. Our results advance research on quantum information processing tasks beyond qubits.

Figure 1

Experimental setup for the MDI verification of quantum steering and randomness generation. It consists of three parts: The source (yellow region) illustrates the preparation of the class of states as per Eq. (7); the pink region named Alice realizes two measurements given in Eqs. (8) and (9) on Alice’s subsystem; the pink region named Bob and Charlie generates the question states $\{{\tau }_{b,j}^T\}$ sent from Charlie and the partial Bell state measurement. In particular, a class of entangled two-qutrit states is encoded in the hybrid of the path and polarization degrees of freedom of photons, and noise is added with a pair of coherence-destroyed photons. Alice’s measurements are implemented via the configuration composed of a series of HWPs, BDs, QWP, and PBS. The question states are encoded in the extra path degree of freedom of the photon, and the measurement projector onto $|{\mathrm{\Phi }}_3⟩$ is realized in the same way as the state preparation process. BD: beam displacer; PBS: polarizing beam splitter; HWP: half-wave plate; DHWP: dichroic half-wave plate; QWP: quart-wave plate; $D$: single photon detector.

Figure 2

Steering parameter $S$ with $p\in [0.6,1]$. Our experimental data for the class of states (7) are marked as blue points and then fitted in the solid blue line. The theoretical prediction of $S_{\mathrm{EPR}}$ for $d=3$ is plotted in the solid red line, while the corresponding bound $S_{\mathrm{LHS}}$ for LHS models is shown in the horizontal black line. For contrast, we also plot the theoretical predictions of $S_{\mathrm{EPR}}$ (dotted red line) and $S_{\mathrm{LHS}}$ (dotted black line) for $d=2$. Note that the minimum value of $p$ for Alice to demonstrate steerability is 0.683 for the theoretical prediction and 0.730 for our fitted data, which are highlighted by two dotted black vertical lines. The error bars are of the order of ${10}^{-3}$, much smaller than the marker size.

Figure 3

Randomness $H_{\min }$ with $p\in [0.6,1]$. The blue dots describe the randomness generated from our observed data; the solid blue curve corresponds to the fitting result, while the solid red curve is the theoretical prediction for states as per Eq. (7). By contrast, the expectation of randomness generation for qubit systems is plotted in the dotted red curve, and its maximal randomness with two settings is one bit (the horizontal black line). The error bars are of the order of ${10}^{-2}$.