Measurement Incompatibility versus Bell Nonlocality: An Approach via Tensor Norms

Measurement incompatibility and quantum nonlocality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements, one characterizing measurement compatibility and the other characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the Clauser-Horne-Shimony-Holt inequality and its variants are the only ones that satisfy it.

Popular Summary

We connect two fundamental notions in quantum mechanics: measurement incompatibility and nonlocality of correlations. In general, two or more quantum measurements cannot be realized simultaneously on a single given quantum system. For example, the simultaneous measurement of the position and the momentum of a quantum particle is prohibited by the famous Heisenberg uncertainty relation. The strange phenomenon of quantum nonlocality addresses a different situation, where some experimental correlations cannot be explained by local hidden variable models. A relation between these two phenomena exists: the presence of nonlocal correlations implies that the measurements performed during the experiment were incompatible.

In our work we study the reverse implication: when does measurement incompatibility guarantee nonlocal correlations? We answer this problem by modeling the two phenomena in the framework of nonlocal games using the mathematical tool of tensor norms. We formulate both incompatibility and nonlocality of correlations using two such tensor norms and compare them. We show that in some specific sense, the famous CHSH Bell inequality for correlations is the only one characterizing incompatibility in some particular setting.

Figure 1

In this work, we relate measurement incompatibility with Bell inequalities using the formalism of tensor norms. Pairwise connections having already been established in the literature, we bring the three concepts together for the first time.

Figure 2

A diagrammatic representation of a quantum measurement apparatus. The device has an input canal and a set of $k$ light-emitting diodes that will turn on when the corresponding outcome is achieved. After the measurement is performed, the particle is destroyed and the apparatus displays the classical outcome (here, 2).

Figure 3

The joint measurement of $A$ and $B$ is simulated by a third measurement $C$, followed by classical postprocessing.

Figure 4

The norm $\Vert A{\Vert }_{M_t^{\prime }}$ for $y\in [-1,1]$ and different values of $t$. The measurements $A$ are ${\sigma }_X$ and $y{\sigma }_Y$, a noisy version of ${\sigma }_Y$. For $t=1$ (the CHSH game), we observe violations (i.e., $\Vert A{\Vert }_{M_t^{\prime }}>\beta (M_t^{\prime })=1$) for every value of $y\in [-1,1]$.

Figure 5

The filled region corresponds to the set of parameters $(y,t)$ for which Alice’s measurements (${\sigma }_X$ and $y{\sigma }_Y$) are Bell nonlocal (for the game $M_t^{\prime }$): $\Vert A{\Vert }_{M_t^{\prime }}>1=\beta (M_t^{\prime })$. Note that for $t>t_{\ast }=(9-4\sqrt{2})/7$, the game $M_t^{\prime }$ does not allow quantum violations when Alice’s measurements are of the form $A=({\sigma }_X,y{\sigma }_Y)$.

Figure 6

The ratio $\Vert A{\Vert }_{M_t^{\prime }}/\Vert A{\Vert }_c$ for $y\in [-1,1]$ and different values of $t$. We note that the ratio is always smaller than 1, except for $t=1$, which is the CHSH game.

Figure 7

The $p$,$q$ region where Alice observe a violation $\Vert A{\Vert }_{G^{\prime }(p,q)}>1$ for different values of $y$.

Figure 8

The norm $\Vert A{\Vert }_{G^{\prime }(p,q)}$ for $y\in [-1,1]$ and different values of $p$ and $q$.

Figure 9

The ratio $\Vert A{\Vert }_{G^{\prime }(p,q)}/\Vert A{\Vert }_c$ for $y\in [-1,1]$ and different values of $p$ and $q$.