Device-independent quantification of measurement incompatibility

Incompatible measurements, i.e., measurements that cannot be simultaneously performed, are necessary to observe nonlocal correlations. It is natural to ask, e.g., how incompatible the measurements have to be to achieve a certain violation of a Bell inequality. In this paper, we provide the direct link between Bell nonlocality and the quantification of measurement incompatibility. This includes quantifiers for both incompatible and genuine-multipartite incompatible measurements. Our method straightforwardly generalizes to include constraints on the system's dimension (semi-device-independent approach) and on projective measurements, providing improved bounds on incompatibility quantifiers, and to include the prepare-and-measure scenario.

Figure 1

Schematic representation of device-independent quantification of measurement incompatibility. By performing measurements on two distant particles and observing the correlations $P(a,b|x,y)$, we are able to estimate several measures of incompatibility among pairs, $(1,2)$, $(1,3)$, $(2,3)$, or triples, $(1,2,3)$, or genuine triplewise incompatibility among $(1,2,3)$.

Figure 2

DI $\mathcal{IR}$ bounds with the MMM method and nonlocal robustness (NLR). Red curve, $\mathcal{IR}$ of Bob's optimal measurements for the tilted-CHSH inequality; blue squares, DI lower bound from the MMM method (second level of the hierarchy); black crosses, lower bound from NLR.

Figure 3

MMMs can also be used to compute lower bounds on genuine triplewise $\mathcal{IR}$ in a DI setting. (a) DI lower bounds on genuine triplewise $\mathcal{IR}$ in the elegant Bell scenario. (b) The black dashed, red dash-dotted, and blue solid curves represent, respectively, DI lower bounds on genuine triplewise $\mathcal{IR}$ in the $I_{3322}$, $I_{3422}^3$, and $I_{3422}^2$ scenarios. The SDP carrying out the computation can be found in Eq. (13).

Figure 4

Quantification of incompatibility robustness for given violation of $I_{3322}$. Blue dash-dotted line, SDI approach for $d=2$; black solid line, DI approach. A similar curve (with a difference of the order of ${10}^{-5}$) corresponds to the SDI calculation with the additional assumption of projective measurements (SDI+proj.) for $d=3,4$ (identical up to numerical precision). Red dashed line, SDI approach with additional assumption of projective measurements for $d=2$.

Figure 5

Comparison between two SDPs computing the bounds. The solid curves are bounds obtained by fixing a Bell violation and minimizing the robustness [cf. Eq. (12)], while the dashed curves are bounds obtained by fixing a robustness and maximizing the Bell violation [cf. Eq. (C5)].

Figure 6

Comparison between lower bounds on $\mathcal{IR}$ in the $I_{3322}$ scenario [60]. The blue solid and black dashed curves represent lower bounds obtained from our method and from the method of the assemblage moment matrices [34], respectively. The local and quantum bounds for the $I_{3322}$ inequality are 0 and around $0.250875561$, respectively [70]. The level of the hierarchy of the semidefinite relaxation used to carry out the computation in both methods is the third level.