Entanglement-free witnessing of quantum incompatibility in a high-dimensional system

Quantum incompatibility, as an important quantum resource, is believed to play a key role in quantum information processing tasks. In order to detect this property, an entanglement-free incompatibility witness is proposed via constructing a combined quantum state discrimination task with prior-measurement and post-measurement information strategies. Here we experimentally implement the witness of the incompatibility of mutually unbiased measurements under different noise levels in a qutrit system. Moreover, we estimate the robustness of incompatibility using collected data. Our work paves the path for understanding quantum incompatibility and exploring the features of the quantum resource.

Figure 1

Experimental setup for quantum incompatible witness. An ultraviolet pulsed laser is used to generate high-quality path-polarization qutrit states in a nonlinear crystal (BBO). The trigger photon goes through two 3-nm filters and one 8-nm filter for the extraordinary and ordinary photon, respectively. The other photon used for state preparation is sent through a half-wave plate (HWP1) which is set at $17.{63}^{\circ }$ and a double polarization beam splitter (DPBS). By changing two HWPs, a quarter-wave plate (QWP), and a phase shifter, twelve MUB states are generated. In the portion of measure the qutrit, DPBS, PBS, HWPs, and QWPs are used to construct the measurement observables. The angles of HWPs and QWPs are chosen to project the state to the eigenstates of the corresponding guessing measurement. The outputs are detected by four silicon avalanche photodiode detectors $D_0,D_1,D_2,D_3$ .

Figure 2

Experimental guessing probabilities with prior- and post-measurement information $P_{\mathrm{guess}}^{\mathrm{prior}}, P_{\mathrm{guess}}^{\mathrm{post}}$ in various witnesses.

Figure 3

Two measurements $(M,N)$ corresponding to the set of $(s_X,s_y)$ in Eq. (6). (a) Compatible and incompatible regions in $A$ and $B$ bases. (b) Compatible and incompatible regions in $C$ and $D$ bases. The measurements in the blue region and grey region are compatible and incompatible, respectively. The measurements above the curves in the top-right corner region are detected by our witnesses.

Figure 4

The lower bound of the robustness of incompatibility versus noises $(s_x,s_y)$ in measurements $M$ and $N$. (a) The robustness of incompatibility in $A$ and $B$ bases. (b) The robustness of incompatibility in $C$ and $D$ bases.

Figure 5

Experimental and theoretical guessing probabilities with two strategies. Experimental results of the guessing possibility from six ensembles are shown in yellow bars and the dash empty bars indicate ideal measurement results for comparison.