Quantum Correlations Are Stronger Than All Nonsignaling Correlations Produced by $n$-Outcome Measurements

We show that, for any $n$, there are $m$-outcome quantum correlations, with $m>n$, which are stronger than any nonsignaling correlation produced from selecting among $n$-outcome measurements. As a consequence, for any $n$, there are $m$-outcome quantum measurements that cannot be constructed by selecting locally from the set of $n$-outcome measurements. This is a property of the set of measurements in quantum theory that is not mandatory for general probabilistic theories. We also show that this prediction can be tested through high-precision Bell-type experiments and identify past experiments providing evidence that some of these strong correlations exist in nature. Finally, we provide a modified version of quantum theory restricted to having at most $n$-outcome quantum measurements.

Figure 1

Illustration of a three-outcome measurement which can be explained as selecting one from three two-outcome measurements. From the outside, the measurement apparatus (represented by the outer box) has three outcomes (represented by three lights of different colors). The state of a physical system tested by the apparatus is described by ${\eta }_{\alpha }$, where $\alpha =1$, 2, 3 is a variable that is hidden to the experimenter but can be read off by the measurement apparatus (illustrated by a robot inside the box using a magnifying glass), without disturbing the state of the system. From the inside, the measurement apparatus works as follows: based on the value of $\alpha$ (here, $\alpha =3$), a corresponding two-outcome measurement $D_{\alpha }$ is selected (as the robot does by operating the switch selecting the measurement $D_3$).