Twenty years of quantum contextuality at USTC

Figure  1.  The Yu-Oh 13-ray appearing in the state-independent proof of contextuality by Yu and Oh^[21]. Left: the geometric representation of the rays in a unit cube. The rays are defined as $y^{-}_{1} =(0,1,-1),\; y_{2} ^{-} =(-1,0,1),\; y_{3} ^{-} =(1,-1,0),\; y_{1} ^{+} =(0,1,1),\; y_{2} ^{+} =(1,0,1),\; y_{3} ^{+} =(1,1,0),\; h_{0} =(1,1,1),\; h_{1} =(-1,1,1),\; h_{2} =(1,-1,1),\; $$ h_{3} =(1,1,-1),\; z_{1} =(1,0,0),\; z_{2} =(0,1,0),\; z_{3} =(0,0,1).$ Right: the orthogonality relationship among the set of rays. Each vertex represents a ray; when two vertices are linked by an edge, the corresponding rays are orthogonal. Figure taken from Ref. [21].

Figure  2.  Contextuality from a Platonic graph. (a) A regular icosahedron is a Platonic solid with 12 vertices and 20 edges. (b) The icosahedron graph (vertices 1–12) is the skeleton of the icosahedron. With the auxiliary vertices 13–16 every vertex belongs to a 4-clique, and the graph’s complement graph has a Lovász orthogonal representation^[151] in dimension 4. (c) The violation of the noncontextuality inequality dual to the icosahedron graph decreases with the linear entropy of a quantum state characterizing the mixedness of the state. Figure taken from Ref. [40].

Figure  3.  First experimental test of contextuality at USTC. Main: experimental setup. A heralded single photon’s path and polarization degrees of freedom encode two qubits. The half-wave plates and polarizing beam splitters inside the two Mach-Zehnder interferometers conducted the first joint path-polarization measurement and that after the interferometers executed the second joint measurement. HWP half-wave plate and PBS polarizing beam splitter. Inset: Experimental result showing event probabilities in accord with the predictions of the noncontextual hidden-variable and quantum theories. Figure adapted from Ref. [23].

Figure  4.  A “standard” experimental setup for testing noncontextuality inequalities containing up to two-point correlations with a photonic qutrit system. To extract the two-point correlation without prematurely destroying the photon, the measurement result of the first observable is registered in the path degree of freedom. The inset shows the experimental violation of the noncontextuality inequality (17) for seven pure states and the maximally mixed state. Figure adapted from Ref. [47].

Figure  5.  Simplification of contextuality experiments. Top: implementing successive measurements poses the main technical challenge on photonic contextuality experiments. Middle: by adopting the graph-theoretical approach to contextuality, the required number of sequential measurements can be reduced to one. Bottom: by assuming the Lüders’ rule, the sequential measurement can be substituted by a destructive measurement and a repreparation procedure, thus completely lifting the requirement of sequential measurements from contextuality experiments at the price of some conceptual disadvantages. Figure taken from Ref. [36].

Figure  6.  A photonic prepare-and-measure setup for testing graph-theoretic noncontextuality inequalities. (a)－(c) With the repreparation procedure, the two-point correlations can be calculated via Eq. (13) and Eq. (19). (d) Experimental results of the contextuality test. (e) Verification of the no-signaling condition. Figure taken from Ref. [38].

Figure  7.  A high-dimensional photonic prepare-and-measure setup, where the quantum information was encoded on the orbital angular momentum degree of freedom.

Figure  8.  Observation of an all-versus-nothing contextuality. (a) Experimental setup. By pumping a nonlinear crystal twice, the two photons received by the two observers became path-polarization hyperentangled. Different apparatuses were devised to measure different path observables. (b) Predictions by noncontextual hidden-variable theory (left) and quantum theory (right) on the probabilities of events in Eq. (23). (c) Experimental results and the quantum prediction are in accord. Figure taken from Ref. [75]

Figure  9.  Experimental setup for observing a compatibility-loophole-free contextuality. Part A implemented state preparation, where maximally entangled qutrits were generated on the photonic path degrees of freedom. Part B implemented qutrit measurements, where the path states of the photons were analyzed. Figure taken from Ref. [87].

Figure  10.  Behaviors of topological contextuality under braiding and local noise. (a) A Bloch sphere illustrating the effect of $ \mathbb{Z}_3 $-parafermion braiding on the encoded qutrit. A geometric phase of $ 2\pi/3 $ is obtained between one and the other two computational bases. (b) The real parts of the theoretical (edge) and experimental (filling) braiding process matrices. (c) Permutation of the magic contextuality witnesses. (d) The violations of magic noncontextuality inequality (32) were almost invariant for the sample states. (e) The real parts of the theoretical (edge) and experimental (filling) local noise process matrices. (f) The dynamics of magic contextuality under hopping noise for a topologically-protected qutrit (blue) and under flip noise for a trivial (not protected) qutrit (orange). Figure adapted from Ref. [107].

Figure  11.  Setup for observation of nonlocality activated by local contextuality. (a) Schematic illustration of the experiment. Alice and Bob share two pairs of Bell states. Alice implements a contextuality test on her qubits and checks the correlations of her observables with Bob’s. (b) Devices for measuring the nine observables in the contextuality test. (c) Experimental result demonstrating the activation of nonlocality from local contextuality. Figure adapted from Ref. [126].