Overview
Bell’s theorem states that the concept of local realism, favoured by Einstein, yields predictions that disagree with those of quantum mechanical theory. Because numerous experiments agree with the predictions of quantum mechanical theory, and show correlations that are, according to Bell, greater than could be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, then the results which are in agreement with quantum mechanical theory appear to evidence superluminal effects, in contradiction to the principle of locality.
The theorem applies to any quantum system of two entangled qubits. The most common examples concern systems of particles that are entangled in spin or polarization.
Following the argument in the Einstein–Podolsky–Rosen (EPR) paradox paper (but using the example of spin, as in David Bohm's version of the EPR argument), Bell considered an experiment in which there are "a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions." The two particles travel away from each other to two distant locations, at which measurements of spin are performed, along axes that are independently chosen. Each measurement yields a result of either spin-up (+) or spin-down (−).
The probability of the same result being obtained at the two locations varies, depending on the relative angles at which the two spin measurements are made, and is subject to some uncertainty for all relative angles other than perfectly parallel alignments (0° or 180°). Bell's theorem thus applies only to the statistical results from many trials of the experiment. Symbolically, the correlation between results for a single pair can be represented as either "+1" for a match (opposite spins), or "−1" for a non-match. While measuring the spin of these entangled particles along parallel axes will always result in opposite (i.e., perfectly anticorrelated) results, measurement at perpendicular directions will have a 50% chance of matching (i.e., will have a 50% probability of an uncorrelated result). These basic cases are illustrated in the table below.
| Same axis | Pair 1 | Pair 2 | Pair 3 | Pair 4 | … | Pair n | |
|---|---|---|---|---|---|---|---|
| Alice, 0° | + | − | − | + | … | + | |
| Bob, 0° | - | + | + | - | … | - | |
| Correlation: ( | +1 | +1 | +1 | +1 | … | +1 | ) / n = +1 |
| Orthogonal axes | Pair 1 | Pair 2 | Pair 3 | Pair 4 | … | Pair n | |
| Alice, 0° | + | − | + | − | … | − | |
| Bob, 90° | − | − | + | + | … | − | |
| Correlation ( | +1 | -1 | -1 | +1 | … | −1 | ) / n = 0 |
With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables would imply a linear variation in the correlation. However, according to quantum mechanical theory, the correlation varies as the cosine of the angle. Experimental results match the curve predicted by quantum mechanics.
Bell achieved his breakthrough by first deriving the results that he posits local realism would necessarily yield. Bell claimed that, without making any assumptions about the specific form of the theory beyond requirements of basic consistency, the mathematical inequality he discovered was clearly at odds with the results (described above) predicted by quantum mechanics and, later, observed experimentally. If correct, Bell's theorem appears to rule out local hidden variables as a viable explanation of quantum mechanics (though it still leaves the door open for non-local hidden variables). Bell concluded:
In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that a theory could not be Lorentz invariant. —Over the years, Bell's theorem has undergone a wide variety of experimental tests. However, various common deficiencies in the testing of the theorem have been identified, including the detection loophole and the communication loophole. Over the years experiments have been gradually improved to better address these loopholes, but no experiment to date has simultaneously fully addressed all of them. However, it is generally considered unreasonable that such an experiment, if conducted, would give results that are inconsistent with the prior experiments. For example, Anthony Leggett has commented:
no single existing experiment has simultaneously blocked all of the so-called ‘‘loopholes’’, each one of those loopholes has been blocked in at least one experiment. Thus, to maintain a local hidden variable theory in the face of the existing experiments would appear to require belief in a very peculiar conspiracy of nature.
To date, Bell's theorem is generally regarded as supported by a substantial body of evidence and is treated as a fundamental principle of physics in mainstream quantum mechanics textbooks.
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