Device-Independent Tests of Structures of Measurement Incompatibility

In contrast with classical physics, in quantum physics some sets of measurements are incompatible in the sense that they cannot be performed simultaneously. Among other applications, incompatibility allows for contextuality and Bell nonlocality. This makes it of crucial importance to develop tools for certifying whether a set of measurements respects a certain structure of incompatibility. Here we show that, for quantum or nonsignaling models, if the measurements employed in a Bell test satisfy a given type of compatibility, then the amount of violation of some specific Bell inequalities becomes limited. Then, we show that correlations arising from local measurements on two-qubit states violate these limits, which rules out in a device-independent way such structures of incompatibility. In particular, we prove that quantum correlations allow for a device-independent demonstration of genuine triplewise incompatibility. Finally, we translate these results into a semidevice-independent Einstein-Podolsky-Rosen-steering scenario.

Figure 1

Examples of incompatibility structures for four measurements. Each node represents a complete measurement (i.e., a complete set of positive-value-operator measure elements), and each hyperedge (represented by a region colored with the same color) contains measurements that are compatible. If a set of measurements are not contained in a hyperedge they are incompatible. The respective incompatibility structures are represented by the following hypergraphs: (a) ${\mathcal{C}}_{\mathrm{A}}=[(1,2)]$; (b) ${\mathcal{C}}_{\mathrm{B}}=[(1,2),(3,4)]$; (c) ${\mathcal{C}}_{\mathrm{C}}=[(1,2),(1,3),(2,4),(3,4)]$; (d) ${\mathcal{C}}_{\mathrm{D}}=[(1,2,3),(2,4)]$.

Figure 2

The set of noisy Pauli measurements $X^{\eta }$, $Y^{\eta }$, $Z^{\eta }$ defined by (2) for $\eta =[(\sqrt{2}+1)/3]$ can be written as a uniform convex combination of Pauli measurements that have a compatible pair (represented in a shaded area). Thus, one can implement these measurements by randomly implementing sets of measurements that are not triplewise incompatible.

Figure 3

Geometrical interpretation of sets of three pairwise and triplewise compatible measurements. Here $L_{123}$ is the standard local set, where all local measurements are compatible. The set $L_{ij}^{\mathrm{NS}}$ consists of probabilities that are nonsignaling and are partially local with respect to $i$ and $j$, i.e., are local when only the measurements $i$ and $j$ are considered on Alice’s side. Analogously, $L_{ij}^Q$ is a set of behaviors that are quantum and partially local with respect to measurements $i$ and $j$.