Scalable noncontextuality inequalities and certification of multiqubit quantum systems

We propose a family of noncontextuality inequalities and show that they can be used for certification of multiqubit quantum systems. Our scheme, unlike those based on nonlocality, does not require spatial separation between the subsystems, yet it makes use of certain compatibility relations between measurements. Moreover, it is scalable in the sense that the number of expectation values that are to be measured to observe contextuality scales polynomially with the number of qubits that are being certified. In a particular case we also show our scheme to be robust to errors and experimental imperfections. Our results seem promising as far as certification of physical setups is concerned in scenarios where spatial separation between the subsystems cannot be guaranteed.

Figure 1

Measurement setup. A contextuality experiment comprises of a preparation device $\mathcal{P}$ that prepares some quantum state $\rho$ which is later measured sequentially by the nondemolishing measurement device with settings ${\mathcal{A}}_i$; each of these measurements yields the $\pm 1$ outcome. Figure from Ref. [33].

Figure 2

Two nonisomorphic graphs with three vertices.

Figure 3

Hypergraph [47] of compatibility of the Kochen-Specker contextuality scenario associated to inequality (20). In this hypergraph, the vertices represent the observables of the scenario and the hyperedges represent the contexts. The red hyperedges are associated to the correlators which enter inequality (20) with $+1$ and the blue to correlators corresponding to the negative sign. Here the colors are conveniently chosen in order to elucidate symmetric properties of the inequality.

Figure 4

Hypergraphs of compatibility of subsets of observables related with two choices of subsets of correlators in Eq. (58). On the left we have the hypergraph associated with the subscripts 1, 2, and 3, and on the right side with the subscripts 1, 2, and 4. These two hypergraphs also represent the compatibility structures of the observables corresponding to the simpler inequality (20) and another such simpler inequality with the observables $A_i$ and $B_j$, with $i,j=1,2,4$, respectively. These two compatibility structures serve as the building blocks to construct inequality (58).