What is Bell's Theorem?

John Bell receiving an Honorary degree at Queen's University Belfast, July 1988.
By Queen's University Belfast (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Question: What is Bell's Theorem?

One of the most curious elements of physics is the principle of quantum entanglement in quantum physics, where two seemingly independent particles appear to be connected to each other in a strange way. This behavior - which was famously debated by Albert Einstein and Niels Bohr - was called "spooky action at a distance" by Einstein. However, physicist John Stewart Bell developed a way of determining whether this "action at a distance" (or non-local behavior, in more physics-like jargon) actually takes place.

What was Bell's Theorem?

Answer: Bell's Theorem was devised by Irish physicist John Stewart Bell (1928-1990) as a means of testing whether or not particles connected through quantum entanglement communicate information faster than the speed of light. Specifically, the theorem says that no theory of local hidden variables can account for all of the predictions of quantum mechanics. Bell proves this theorem through the creation of Bell inequalities, which are shown by experiment to be violated in quantum physics systems, thus proving that some idea at the heart of local hidden variables theories has to be false. The property which usually takes the fall is locality - the idea that no physical effects not move faster than the speed of light.

Quantum Entanglement

In a situation where you have two particles, A and B, which are connected through quantum entanglement, then the properties of A and B are correlated.

For example, the spin of A may be 1/2 and the spin of B may be -1/2, or vice versa. Quantum physics tells us that until a measurement is made, these particles are in a superposition of possible states. The spin of A is both 1/2 and -1/2. (See our article on the Schroedinger's Cat thought experiment for more on this idea.

This particular example with particles A and B is a variant of the Einstein-Podolsky-Rosen paradox, often called the EPR Paradox.)

However, once you measure the spin of A, you know for sure the value of B's spin without ever having to measure it directly. (If A has spin 1/2, then B's spin has to be -1/2. If A has spin -1/2, then B's spin has to be 1/2. There are no other alternatives.) The riddle at the heart of Bell's Theorem is how that information gets communicated from particle A to particle B.

Bell's Theorem at Work

John Stewart Bell originally proposed the idea for Bell's Theorem in his 1964 paper "On the Einstein Podolsky Rosen paradox." In his analysis, he derived formulas called the Bell inequalities, which are probabilistic statements about how often the spin of particle A and particle B should correlate with each other if normal probability (as opposed to quantum entanglement) were working. These Bell inequalities are violated by quantum physics experiments, which means that one of his basic assumptions had to be false, and there were only two assumptions that fit the bill - either physical reality or locality was failing.

To understand what this means, go back to the experiment described above.

You measure particle A's spin. There are two situations that could be the result - either particle B immediately has the opposite spin, or particle B is still in a superposition of states.

If particle B is affected immediately by the measurement of particle A, then this means that the assumption of locality is violated. In other words, somehow a "message" got from particle A to particle B instantaneously, even though they can be separated by a great distance. This would mean that quantum mechanics displays the property of non-locality.

If this instantaneous "message" (i.e., non-locality) doesn't take place, then the only other option is that particle B is still in a superposition of states. The measurement of particle B's spin should therefore be completely independent of the measurement of particle A, and the Bell inequalities represent the percent of the time when the spins of A and B should be correlated in this situation.

Experiments have overwhelmingly shown that the Bell inequalities are violated. The most common interpretation of this result is that the "message" between A and B is instantaneous. (The alternative would be to invalidate the physical reality of B's spin.) Therefore, quantum mechanics seems to display non-locality.

Note: This non-locality in quantum mechanics only relates to the specific information that is entangled between the two particles - the spin in the above example. The measurement of A cannot be used to instantly transmit any sort of other information to B at great distances, and no one observing B will be able to tell independently whether or not A was measured. Under the vast majority of interpretations by respected physicists, this does not allow communication faster than the speed of light.

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Jones, Andrew Zimmerman. "What is Bell's Theorem?" ThoughtCo, Jul. 25, 2016, thoughtco.com/what-is-bells-theorem-2699344. Jones, Andrew Zimmerman. (2016, July 25). What is Bell's Theorem? Retrieved from https://www.thoughtco.com/what-is-bells-theorem-2699344 Jones, Andrew Zimmerman. "What is Bell's Theorem?" ThoughtCo. https://www.thoughtco.com/what-is-bells-theorem-2699344 (accessed September 21, 2017).