Simple method for experimentally testing any form of quantum contextuality

Contextuality provides a unifying paradigm for nonclassical aspects of quantum probabilities and resources of quantum information. Unfortunately, most forms of quantum contextuality remain experimentally unexplored due to the difficulty of performing sequences of projective measurements on individual quantum systems. Here we show that two-point correlations between binary compatible observables are sufficient to reveal any form of contextuality. This allows us to design simple experiments that are more robust against imperfections and easier to analyze, thus opening the door for observing interesting forms of contextuality, including those requiring quantum systems of high dimensions. In addition, it allows us to connect contextuality to communication complexity scenarios and reformulate a recent result relating contextuality and quantum computation.

Figure 1

(a) A typical contextuality experiment requires preparing individual quantum systems in a quantum state $|\psi \rangle$ and performing a sequence of projective measurements on each of them. (b) Contextuality experiments can be substantially simplified by exploiting that any form of contextuality can be tested by measuring sequences of only two compatible binary observables. The problem is that in most quantum systems sequential projective measurements are unfeasible. In these cases, a possible test of contextuality (not without drawbacks) is one based on the scheme in panel (c). It requires replacing the projective measurement of $i$ in panel (b) by a demolition measurement of $i$ followed, in those trials with outcome $i=1$, by a preparation of the eigenstate of $i$ with eigenvalue 1 or, in those trials with outcome $i=0$, by a preparation of the quantum state predicted by Lüders's rule.

Figure 2

Simplest nontrivial example of how the graph $G$ is related to the graph $G^{\prime }$. (a) For any graph $G$ with vertex set $V\left(G\right)$, there is a NC inequality of the type $S={\sum }_{i\in V}P(1,0,...,0|i,i_1,...,i_{n\left(i\right)})\stackrel{\mathrm{NCHVTs}}{\le }\alpha \left(G\right)$, which is violated by QT up to $\vartheta \left(G\right)$; the set ${\left\{i_j\right\}}_{j=1}^{n\left(i\right)}$ contains the observables corresponding to vertices adjacent to vertex $i$. The probabilities in this inequality are well defined both in NCHVTs and in QT but, in general, involve incompatible tests. This is a problem for designing experiments. (b) By replacing the probabilities $P(1,0,...,0|i,i_1,...,i_{n\left(i\right)})$ in $G$ by $P\left(1\right|i)$ and every edge $(i,j)\in G$ by the vertices corresponding to $P(0,0|i,j), P(0,1|i,j)$, and $P(1,0|i,j)$ and adding edges between vertices corresponding to exclusive events, we construct a new graph $G^{\prime }$ which is associated to a new NC inequality, given in Eq. (1), which has the same bounds for NCHVTs and QT that the NC inequality associated to $G$, but involves only compatible observables and can be tested by measuring only two-point correlations.