Measurement compatibility in Bell nonlocality tests

Incompatibility of observables, or measurements, is one of the key features of quantum mechanics, related, among other concepts, to Heisenberg's uncertainty relations and Bell nonlocality. In this manuscript we show, however, that even though incompatible measurements are necessary for the violation of any Bell inequality, some relevant Bell-like inequalities may be obtained if compatibility relations are assumed between the local measurements of one (or more) of the parties. Hence, compatibility of measurements is not necessarily a drawback and may, however, be useful for the detection of Bell nonlocality and device-independent certification of entanglement.

Figure 1

Upper panel: a standard Bell nonlocality scenario. In each round of the experiment, party A (B) chooses measurement $x$ ($y$) to perform on its respective subsystem, obtaining outcome $a$ ($b$). The experiment is described by the conditional probabilities $p(a,b|x,y)$. Lower panel: a Bell nonlocality scenario where party B is able to perform compatible measurements. Now, in each round, party B chooses two (or more, according to the context) measurements to be jointly performed, $y$ and $y^{\prime }$, obtaining outcomes $b$ and $b^{\prime }$, respectively. The experiment is described by the conditional probabilities $p(a,b,b^{\prime }|x,y,y^{\prime })$; defining $\mathsf{y}=(y,y^{\prime })$ and $\mathsf{b}=(b,b^{\prime })$, the same conditional probabilities can be written as $p(a,\mathsf{b}|x,\mathsf{y})$.

Figure 2

For states $\rho \left(\alpha ,w\right)$ defined in Eq. (17), the plot shows the critical parameter $w$, as a function of $\alpha$, above which inequality Eq. (15) is violated (red circles), above which the CHSH inequality is violated (black squares), and above which the $I_{3322}$ inequality is violated (blue triangles). The points for inequalities Eq. (15) and $I_{3322}$ were obtained via a seesaw optimization, and are, hence, upper bounds on the actual critical points. The points for the CHSH are exact, obtained by means of Horodecki's necessary and sufficient criterium for violation of the CHSH inequality by two-qubit states [35]. Inequality Eq. (15) is denoted $I_{\#15}$ for consistency with the Appendices and the data related to the red points, which are available at Ref. [37].

Figure 3

For states $\sigma (\alpha ,w)$ defined in Eq. (19), the plot shows the critical parameter $w$, as a function of $\alpha$, above which inequality Eq. (15) is violated (red circles), and above which the CHSH inequality is violated (black squares). The points for inequality Eq. (15) were obtained via a seesaw optimization, and are, hence, upper bounds on the actual critical points. The points for the CHSH are exact, obtained by means of Horodecki's necessary and sufficient criterium for violation of the CHSH inequality by two-qubit states [35]. As in Fig. 2, inequality Eq. (15) is denoted ${\mathrm{I}}_{\#15}$ for consistency with the Appendices and the data related to the red points, which are available at Ref. [37].