Microscopic experiment. If Alice and Bob repeat an experiment many times, each time applying random interactions X and Y , they can compare their measurement outcomes, make statistics and obtain a set of probabilities P(a,b), the probability of measuring clicks in detectors D(a),D(b) when Alice applies an interaction X(a) and Bob applies an interaction Y (b). From now on, we will assume that the no-signalling condition holds, i.e. that, for any X,X′ with X≠X′, , and, for any Y,Y ′ with Y ≠Y ′, . These two conditions just assert that Alice’s choice of measurement setting cannot affect Bob’s statistics and vice versa. Also, the set of marginal probability distributions P(a,b), in general, will not admit a local hidden variable model. This means that, in some cases, there will not exist a joint probability distribution for all 2s possible measurements P(c1,…,c2s), such that .
Macroscopic experiment. Alice and Bob receive two particle beams, interact with them collectively and then take a record of the intensities measured on each detector. A macroscopic experiment implies restrictions on the physical states to be measured (N identical and independent copies of a microscopic state), on the possible interactions to be performed (identical microscopic interactions over all the particles of each beam) and on the resolution of the detectors used (able to measure intensity fluctuations of order , but unable to resolve individual particles).
Wiring. Example of deterministic wiring between two physical systems: the effective measurement X (Y) of Alice’s (Bob’s) is applied over the first system, giving an outcome a1 (b1), while the interaction to be applied over the second subsystem is a function of both X and a1 (Y and b1). Labelling the second outcome by a2 (b2), the effective outcome of the whole scheme is a function of a1,a2,X (b1,b2,Y). Note that, by definition, wiring is local, i.e. it does not require communication between Alice and Bob.
Departure from quantum mechanics. As for the CHSH parameter, the Collins–Gisin–Linden–Massard–Popescu (CGLMP) parameters (Collins et al. 2002) are linear functions Sd on the probabilities P(a,b), to be applied over scenarios with s=2 and arbitrary d, and, as for the CHSH parameter, the maximum possible value of these functions under the assumption of locality is 2. The plot above (filled squares, macroscopic locality; filled triangles, quantum mechanics) shows the difference between the maximal value of Sd inside Q, as well as inside Q1, as a function of d (Navascués et al. 2008). At first glance, there seems to be room for a wide variety of macroscopically local theories other than quantum mechanics. This suggests that, if our Universe happened to be described by one of these, we might be able to experimentally falsify quantum mechanics with a sufficient number of detectors.