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Bell's Theorem (Quantum vs Classical)

In 1935, Einstein, Podolsky, and Rosen proposed a paradox (hence EPR-paradox) as a thought experiment ("gedankenexperiment" in German) in which the particles can have both their position and momentum accurately measured in violation of the uncertainty principle
EPR Paradox pq > ; it also violates the principle of locality (signal cannot travels faster than the speed of light) in special relativity (see Figure 01). They argued that "elements of reality" such as momentum and position are deterministic properties of particles although it is hidden in quantum theory which endows them with probability only. Hence quantum theory is an incomplete theory, there is no need to invoke "non-locality" and also can do away with the "uncertainty principle" if the hidden cause is included in the theory.

Figure 01 EPR Paradox [view large image]

The thought experiment became real with the discovery of entanglement in 1967. Entanglement occurs for example, in the decay of
Entanglement the pi meson into an electron-positron pair (Figure 02). Since the spin for the pi meson is 0, the spin for the electron-positron pair must be opposite according to the conservation of angular momentum. Therefore, no matter how far apart are the members of this pair, if the spin is flipped for one of the member, the spin for the other member will also be flipped to the opposite at precisely the same moment. This non-local influence (objects can influence each other only locally according to classical physics) occur instantaneously, as if some form of communication, which Einstein called a "spooky action at a distance", operates not just faster than the speed of light, but infinitely fast.

Figure 02 Entanglement [view large image]

Figure 02 demonstrates the origin of entanglement and its final collapse by measurement. The observers in the names of Alice and Bob are modern inventions.

Then a seminal paper by John Bell in 1964 shows that if EPR were correct, the results found by two widely separated detectors measuring certain properties of the two entangled photons (or particles) (such as the polarization direction or spin orientation about various randomly chosen detecting angle) would have to stay inside certain range - this is known as the Bell's theorem, or Bell's inequality. Starting in early 1970s, technology has improved significantly to enable the required experiments for resolving the paradox. It culminated in the early 1980s, when the Aspect experiment firmly established that measurements from the two detectors could be outside the range, i.e., violating the Bell's inequality. Quantum mechanics survived the test and entanglement will stay with us into the quantum computing age.


Bell's Theorem for Dummies An equilvalent form of the Bell's inequality asserts that if EPR were correct, the results found by two widely separated detectors measuring certain properties of 2 entangled particles would have to agree (match) more than 50% of the time. Figure 08 provides a very specific example to illustrate how underlying pre-arrangement (hidden variables) can bump up the chance of QM ramdon match.

Figure 08 Bell's Theorem for Dummies [view large image]

This is Brian Greene's analogy also known as "Bell's Theorem for Dummies".


The tests on the Bell's theorem have at least two loopholes until 2015 when new technology allows experiment to skirt such problems. One problem is known as the "detection loophole" with the lost up to 80% of the entangled qbits (in the forms of photons) and it is not sure that the remaining 20% is the true representative. The use of closely spacing atoms may retain more qbits for the measurements, but it could be masked by under speed of light communication, and it thus earns the name as "communication loophole". The 2015
Bell Test, 2015 experiment at Delft University solves both problems as shown in Figure 09. It produces entanglement of two electrons inside two diamonds respectively (separated by 1.3 km - enough to close the communication loophole and with no lost of entangled qbits) via the entanglement of the photons emitted by each. The occurrence is not very often - just a few per hour. Eventually, 245 measurements were taken to show that the standard quantum view is valid. Difficulty of the experiment produced a p-value of only 4% - a statistical significance just passes the usual 5% and is much shorter than the 1/106 standard for experiment in physics. Anyway, this experiment also guarantees the security in "quantum cryptography", which may be hacked through the loopholes.

Figure 09 Bell Test, 2015

See more detail in "Quantum 'spookiness' passes toughest test yet".


One argument for "hidden variables" suggests that the hidden links were established long time ago; it is there already before the measurement, free-will or random number generator in the measuring process notwithstanding. A 2017 experiment using star light (red, or blue) to trigger the measurement shows that quantum entanglement comes with the "spooky action at a distance" property, i.e., no "hidden variables" to link the pair as far back as 600 years ago when the star light was emitted .

Finally, see "Bell Test Experiments" over the years.