What is **reality**? Is there a mathematically rigorous way to define it? Following the German philosopher Immanuel Kant, we can at least start by vaguely identifying two different kinds of reality: anything that we come across a posteriori as a result of an observation, so-called **phenomenon**, and *anything-in-itself* a priori with respect to our observation, so-called **noumenon**. In the following we will be interested in the latter form of reality and ask the question: is it there even if we do not observe it? In other (Einstein’s) words: is the moon there when no one looks? Of course many other questions immediately follow, such as: how can we ever prove that there is no *reality-in-itself* if we are by definition not allowed to observe it? Can we resort to the phenomenon to prove the nonexistence of the noumenon?

On one hand, in our everyday experience, which deals with *macroscopic* objects and thus fits within the **classical realm**, there is no evidence of the fact that such objects have no reality per se. On the other hand, amazingly enough, some experiments have very recently demonstrated beyond any major loop-hole that at the *subatomic scale*, which lies comfortably within the **quantum realm**, there could be no reality-in-itself if the following very natural condition governed the quantum world: if two events are too close in time and far in space in such a way that not even the light can make it to connect them (so-called **spatially separated** events), then they cannot be in a causal relationship. The latter condition goes under the name of **locality**. So, overall, the crazy quantum wonderland is free from either reality-in-itself or locality or even both!

In order to get an idea of how these experiments managed to prove that at the microscopic scale nature fails to be jointly local and realistic, let us first familiarise a bit more with the concept of local realism by considering the following thought experiment, taken from the inspiring Michel Le Bellac’s Quantum Physics book.

Two travellers *A* and *B*, each carrying a suitcase, depart in opposite directions from the same point *O*. The suitcases of the travellers satisfy the following properties, as also illustrated in the figure below:

- they are circles divided into infinitesimal angular sectors labelled by the corresponding orientations:
*a*,*b*, …, etc.; - each angular sector contains either the result (+) or (-) in such a way that two angular sectors that do not belong to the same suitcase but are labelled by the same orientation contain opposite results.

The travellers have picked up their closed suitcases at random at the starting point *O* and do not know what results are inside.

Eventually, the two travellers are checked by Alice and Bob. Alice opens the angular sector of the suitcase of the traveller *A* labelled by the orientation *a* while Bob opens the angular sector of the suitcase of the traveller *B* labelled by the orientation *b* in such a way that their observations are *spatially separated* in the aforementioned sense.

Then they repeat this experiment several times. What happens is that both Alice and Bob observe a perfect random series of (+) and (-), even though Alice always opens the angular sector of the suitcase of the traveller *A* labelled by the orientation *a* and Bob always opens the angular sector of the suitcase of the traveller *B* labelled by the orientation *b*. This is due to the fact that, at every repetition of the experiment, the two travellers pick up their closed suitcases at random.

After the above series of experiments, Alice and Bob meet, compare their results and calculate the quantity *P(a=+,b=+)*, that is, the joint probability that both the angular sector *a* of the suitcase carried by *A* and the angular sector *b* of the suitcase carried by *B* contain a (+). They calculate this probability by simply dividing the number of times that they both got (+) in a same repetition by the number of repetitions of the experiment. For example, if *a=b*, we have that when Alice and Bob, after the above series of experiments, meet and compare their results they see that there is a perfect anti-correlation between the outcomes of their observations, in the sense that any time Alice has found, e.g., the result (+), in the same repetition of the experiment Bob has found the value (-), and viceversa, so that they would get *P(a=+,a=+)=0*.

This experiment is local and realistic if the following two conditions jointly apply:

*Locality*: the outcome of Alice’s observation, say (+), only reveals a piece of information already stored in the local region of space-time associated with the suitcase carried by*B*: the opposite result, (-), must be in the angular sector labelled by the orientation*a*of the latter suitcase. The correlations between the two suitcases were introduced at the time of departure and reappear as classical correlations between the results of the observations of Alice and Bob. In other words, within this classical experiment, the opening of the suitcase of the traveller*A*by Alice does not disturb the suitcase of the traveller*B*whatsoever, but it determines the result of Bob if he had opened the angular sector of the suitcase of the traveller*B*labelled by the orientation*a*. The sign contained in the latter angular sector existed before the suitcase of*A*was opened by Alice. An analogous condition holds for Bob;*Reality*: even though, in every repetition of the experiment, Bob is allowed to open only the angular sector*b*of the suitcase carried by*B*, this suitcase still has a well defined result in any other angular sector*b’*different from*b*, as illustrated in the figure above. In other words the suitcase carried by*B*has simultaneous physical reality in any of its angular sectors, regardless of whether they are opened (*phenomenon*) or not (*noumenon*). An analogous condition holds for the suitcase carried by*A*.

By jointly making these two assumptions, we can easily obtain a quantitative theoretical prediction on the joint probability of Alice and Bob both getting two pluses as follows:

*P(a=+,b=+) + P(b=+,c=+) ≤ P(a=+,c=+),*

where *c* represents another possible orientation. If such theoretical prediction is found to be violated by Alice and Bob, then the joint local realism hypothesis underlying it is wrong. The above experimental local realism tester was discovered by the Northern Irish physicist John Stewart Bell in 1964 and represented the turning point from a philosophical to a concrete scientific debate regarding the local realistic nature of our universe. Such inequality indeed goes under the name of Bell inequality.

Now by using two suitcases the above inequality can never be violated and so there is no confutation of the local realism hypothesis in the classical realm. On the other hand, what happens if we replace the two macroscopic suitcases with two microscopic electrons travelling towards Alice and Bob? First of all, what do Alice and Bob observe of the received electrons? They measure their spin, which basically is a property of the electron that has infinitely many components in all possible directions and can be measured only in a given direction per time. Moreover, the result of such measurement is dichotomic, i.e., it can be either a (+) or a (-). Therefore, if the spins of the two electrons are in a so-called Einsten-Podolsky-Rosen state, which is perfectly anti-correlated, it turns out that the local realistic way to model them is exactly as the one that we have just used in Le Bellac’s thought experiment for the two suitcases so that exactly the same Bell inequality would apply to such electrons if they were governed by local realism. Well, some experiments have just demonstrated that two electrons in an EPR state violate the above inequality, thus proving that local realism cannot govern the quantum realm!

Nice article. The inequality involving joint probabilities, reproduced from Bellac’s book, is exactly equivalent to the original local realist inequality derived by Wigner in 1970 (cf. http://scitation.aip.org/content/aapt/journal/ajp/38/8/10.1119/1.1976526) considering joint outcomes on the singlet state (which mimics the perfect anticorrelation property assumed for the two suitcases in Bellac’s thought experiment). This can therefore be looked upon as a bonafide Bell inequality exclusive to the singlet (EPR-Bohm) state. However, for suitable choices of the detector settings, violations to this local realist constraint are observed for the singlet. It also makes for a pertinent point here that the usual Bell-CHSH inequality reduces to Wigner’s inequality for the singlet state when you assign a common setting to the two local detectors: say, a and c at one location, and c and b at another, with c being the common setting).

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