Bell's Theorem - Importance of The Theorem

Importance of The Theorem

Bell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox, has been called, on the assumption that the theory is correct, "the most profound in science". Perhaps of equal importance is Bell's deliberate effort to encourage and bring legitimacy to work on the completeness issues, which had fallen into disrepute. Later in his life, Bell expressed his hope that such work would "continue to inspire those who suspect that what is proved by the impossibility proofs is lack of imagination."

The title of Bell's seminal article refers to the famous paper by Einstein, Podolsky and Rosen that challenged the completeness of quantum mechanics. In his paper, Bell started from the same two assumptions as did EPR, namely (i) reality (that microscopic objects have real properties determining the outcomes of quantum mechanical measurements), and (ii) locality (that reality in one location is not influenced by measurements performed simultaneously at a distant location). Bell was able to derive from those two assumptions an important result, namely Bell's inequality, implying that at least one of the assumptions must be false.

In two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, liable to be experimentally tested, thus raising the possibility of testing the local realism hypothesis. Limitations on such tests to date are noted below. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theorem was later generalized to stochastic theories as well, and it was also realised that the theorem is not so much about hidden variables as about the outcomes of measurements which could have been done instead of the one actually performed. Existence of these variables is called the assumption of realism, or the assumption of counterfactual definiteness.

After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two hypothetical observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It is the conclusion of EPR that once Alice measures spin in one direction (e.g. on the x axis), Bob's measurement in that direction is determined with certainty, as being the opposite outcome to that of Alice, whereas immediately before Alice's measurement Bob's outcome was only statistically determined (i.e., was only a probability, not a certainty); thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly.

In QM, predictions are formulated in terms of probabilities — for example, the probability that an electron will be detected in a particular place, or the probability that its spin is up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness is its inability to predict those values precisely. The possibility existed that some unknown theory, such as a hidden variables theory, might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilities predicted by QM. If such a hidden variables theory exists, then because the hidden variables are not described by QM the latter would be an incomplete theory.

Two assumptions drove the desire to find a local realist theory:

1. Objects have a definite state that determines the values of all other measurable properties, such as position and momentum.
2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (in consequence of special relativity). Thus if observers are sufficiently far apart, a measurement made by one can have no effect on a measurement made by the other.

In the form of local realism used by Bell, the predictions of the theory result from the application of classical probability theory to an underlying parameter space. By a simple argument based on classical probability, he showed that correlations between measurements are bounded in a way that is violated by QM.

Bell's theorem seemed to put an end to local realism. This is because, if the theorem is correct, then either quantum mechanics or local realism is wrong, as they are mutually exclusive. The paper noted that "it requires little imagination to envisage the experiments involved actually being made", to determine which of them is correct. It took many years and many improvements in technology to perform tests along the lines Bell envisaged. The tests are, in theory, capable of showing whether local hidden variable theories as envisaged by Bell accurately predict experimental results. The tests are not capable of determining whether Bell has accurately described all local hidden variable theories.

The Bell test experiments have been interpreted as showing that the Bell inequalities are violated in favour of QM. The no-communication theorem shows that the observers cannot use the effect to communicate (classical) information to each other faster than the speed of light, but the ‘fair sampling’ and ‘no enhancement’ assumptions require more careful consideration (below). That interpretation follows not from any clear demonstration of super-luminal communication in the tests themselves, but solely from Bell's theory that the correctness of the quantum predictions necessarily precludes any local hidden-variable theory. If that theoretical contention is not correct, then the "tests" of Bell's theory to date do not show anything either way about the local or non-local nature of the phenomena.