It was Einstein who first set up a standard to prove this. Einstein was so convinced that Realism was true, he concluded that Quantum Mechanics must be faulty. His biggest complaint was that QM seems to violate Locality, or “The Principle of Local Action.” Realism does not allow the possibility of “spooky action at a distance” as Einstein called it:
For the relative independence of spatially distant things (A and B), this idea is characteristic: an influence on A has no immediate effect on B; this is known as the “principle of local action”. - Einstein, Albert (1948). "Quanten-Mechanik Und Wirklichkeit" [Quantum Mechanics and Reality].
Which is corroborated by Newton’s conception of reality as well:
It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact… so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. —Isaac Newton, Letters to Bentley, 1692/3
And so what is known as the famous EPR paper was born. http://www.drchinese.com/David/EPR.pdf
Written in 1935, it assumes in it’s very first paragraph that we operate in an objective reality, and pointing out that QM seems to violate “The Principle of Locality,” concluded that Quantum Mechanics was necessarily incomplete, rather than that we do not in fact operate in an objective reality. For Einstein and his collaborators, if it was between Realism and Quantum Mechanics, they picked Realism. So deeply rooted is the assumption of Realism in the minds of men.
Then came John Bell in 1964 with his paper titled, “On The Einstein Podolsky Rosen Paradox.” http://www.drchinese.com/David/Bell_Compact.pdf
Bell proposed that either Einstein was correct and QM was incomplete and Locality was not actually being violated, or Einstein was wrong and QM was right, but locality is not a true principle of our world. His proposition was that if Einstein was right and QM was wrong, then there must be “Hidden Variables” not described by QM that are causing the apparent interaction at a distance. With Bell’s Theorem, it became possible to experimentally test whether or not Local Realism was actually being violated. If all of the “Hidden Variables” were eliminated as possible explanations of the “spooky action at a distance” then we would have conclusive proof that Local Realism is false.
Since his proposal, several dozen experiments have been performed to test Bell’s Inequalities. These include: Holt and Pipkin (1973), Lamehi-Rachti and Mittig (1976), Freedman and Clauser (1976), Fry and Thompson (1976), Kasday-Ullman-Wu, Faraci-Gutkowski-Notarigo-Pennisi, Wilson-Lowe-Butt, Bruno-d’Agostino-Maroni, Aspect-Grangier-Roger (1982), Aspect-Dalibard-Roger (1982), Shih and Alley, Ou and Mandel, Rarity and Tapster, Zeilinger (1999), Go (2003) and more. All of which confirmed that Quantum Mechanics was not incomplete as Einstein had suggested, and because Bell’s Inequalities are violated, Local Realism is false.
In 2003 A. J. Leggett wrote a paper titled, “Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem.” http://people.isy.liu.se/icg/jalar/kurser/QF/assignments/Leggett2003.pdf
In the opening paragraph he states:
Bell’s celebrated theorem states that, in a situation like that considered by Einstein et al., which involves the correlation of measurements on two spatially separated systems which have interacted in the past, no local hidden-variable theory (or more generally, no objective local theory) can predict experimental results identical to those given by standard quantum mechanics. Over the past thirty years a very large number of experiments have been conducted with the aim of testing the predictions of quantum mechanics against those of local hidden-variable theories, and while to the best of my knowledge no single existing experiment has simultaneously blocked all of the so-called “loopholes” (detector efficiency, random choice of setting, etc.) each one of those loopholes has been blocked in at least one experiment (cf., e.g., Weihs et al.). Thus, to maintain a local hidden-variable theory in the face of existing experiments would appear to require belief in a very peculiar conspiracy of nature. - A. G. Leggett
In the very same paper, Leggett went on to describe his own “Inequalities”, this time of the non-local variety. Bell set out to test for Local Hidden-Variables, whereas Leggett set out to test Non-Local Hidden-variables. In other words, if Bell’s Inequality was violated this would rule out “Local Realism”, and if Leggett’s Inequality was violated this would rule out “Non-Local Realism.” If both were violated this would rule out Realism altogether.
In his paper, Leggett describes the purpose behind this in the paragraph:
The reader might well ask why the whole subject of non-local hidden-variable theories is of any interest. In my view, the point of considering such theories is not so much that they are in themselves a particularly plausible picture of physical reality, but that by investigating their consequences one may attain a deeper insight into the nature of the quantum mechanical “weirdness” which Bell’s Theorem exposes. In particular, I believe that the results of the present investigation provide quantitative backing for a point of view which I believe is by now certainly well accepted at the qualitative level, namely that the incompatibility of the predictions of objective local theories with those of quantum mechanics has relatively little to do with locality and much to do with objectivity. - A. G. Leggett
In essence, Bell set out to disprove Locality as a necessary aspect of objective reality, and thus disproving Local Realism but leaving Non-Local Realism in it’s place, while Leggett set out to go a step further and disprove objectivity itself, ruling out both Local and Non-Local Realism altogether. Realism being that reality is objective and mind-independent, and Idealism committed to reality being subjective and mind-dependent, if Idealism is true we can predict that both Bell’s and Leggett’s hidden-variables will not suffice to keep Realism afloat.
The experiment was successfully completed in 2007 by Anton Zeilinger and his team and described in his paper titled, “An Experimental Test of Non-Local Realism”. http://arxiv.org/pdf/0704.2529.pdf
The following is the complete abstract of that paper. Please read it carefully keeping in mind the above information which lead up to this experiment:
Most working scientists hold fast to the concept of 'realism' - a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of 'spooky' actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.
Just as Idealism predicts, it is not just Locality, but Realism itself that should be abandoned. I highly encourage all readers of this post to read this paper in particular as it does a wonderful job of describing the situation in an accessible way. Here are some highlights from the paper:
Bell’s Theorem proves that all hidden-variable theories based on the joint assumption of locality and realism are at variance with the predictions of quantum physics. Locality prohibits any influences between events in space-like separated regions, while realism claims that all measurement outcomes depend on pre-existing properties of objects that are independent of the measurement. The more refined versions of Bell’s theorem by Clauser, Horne, Shimony and Holt and by Clauser and Horne start from the assumptions of local realism and result in inequalities for a set of statistical correlations (expectation values), which must be satisfied by all local realistic hidden-variable theories. The inequalities are violated by quantum mechanical predictions. Greenberger, Horne and Zeilinger showed that already perfect correlations of systems with at least three particles are inconsistent with these assumptions. So far, all experiments motivated by these theorems are in full agreement with quantum predictions.... Therefore it is reasonable to consider the violation of local realism a well established fact.
The logical conclusion one can draw from the violation of local realism is that at least one of it’s assumptions fails. Specifically, either locality or realism or both cannot provide a foundational basis for quantum theory.... It is therefore important to ask whether incompatibility theorems similar to Bell’s can be found in which at least one of these concepts is relaxed. Our work addresses a broad class of non-local hidden-variable theories that are based on a very plausible type of realism and that provide an explanation for all existing Bell-type experiments. Nevertheless we demonstrate, both in theory and experiment, their conflict with quantum predictions and observed measurement data. Following the recent approach of Leggett, who introduced the class of non-local models and formulated an incompatibility theorem, we have analysed its assumptions and derived an inequality valid for such theories that can be experimentally tested. In addition, the experiments allow for a simultaneous test of all local hidden-variable models - that is, the measurement data can neither be explained by a local realistic model nor by the considered class of non-local models.
The theories are based on the following assumptions: (1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism);....Nevertheless, we will show that all models based on assumptions (1)-(3) are at variance with other quantum predictions.
We believe that our results lend strong support to the view that any future extension of quantum theory that is in agreement with experiments must abandon certain features of realistic descriptions.
Realism is the assumption that physical reality exists independent of observation. The violation of Bell’s Inequalities ruled out Local Realism, and the violation of Leggett’s Inequalities ruled out Non-Local Realism, therefore Realism in any form has been conclusively ruled out.
The completion of this experiment and it’s conclusions are predicted by an Idealist Model of reality which is committed to observation-dependence and the falsity of all conceptions of Realism. After this paper was released, it spawned two very important articles. One in Nature titled “Physicists bid farewell to reality?” http://www.nature.com/news/2007/070416/full/news070416-9.html
, and the other in Physics World titled, “Quantum physics says goodbye to reality.” http://physicsworld.com/cws/article/news/2007/apr/20/quantum-physics-says-goodbye-to-reality
Both released in 2007 immediately following the publication of the paper described and quoted above. I highly recommend reading both articles.