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Setup of the EPR experiment

The following is a simplified description of the EPR scenario, developed by Bohm and Wigner. We follow the approach in Sakurai (1994).

Alice and Bob are two spatially separated observers. Between them is an apparatus that continuously produces pairs of electrons. One electron in each pair is sent towards Alice, and the other towards Bob. The setup is shown below:

The electron pairs are specially prepared so that if both observers measure the spin of their electron along the same axis, then they will always get opposite results. For example, suppose Alice and Bob both measure the z-component of the spins that they receive. According to quantum mechanics, each of Alice's measurements will give either the value +1/2 or -1/2, with equal probability. For each result of +1/2 obtained by Alice, Bob's result will inevitably be -1/2, and vice versa.

Mathematically, the state of each two-electron composite system can be described by the state vector

.

Each ket is labelled by the direction in which the electron spin points. The above state is known as a spin singlet. The z-component of the spin corresponds to the operator (1/2)σz, where σz is the third Pauli matrix. (The quantum mechanics of spin is discussed in the article spin (physics).)

Hidden variables

It is possible to explain this phenomenon without resorting to quantum mechanics. Suppose our electron-producing apparatus assigns a parameter, known as a hidden variable, to each electron. It labels one electron "spin +1/2", and the other "spin -1/2". The choice of which of the two electrons to send to Alice is decided by some classical random process. Thus, whenever Alice measures the z-component spin and finds that it is +1/2, Bob will measure -1/2, simply because that is the label assigned to his electron. This reproduces the effects of quantum mechanics, while preserving the locality principle.

The appeal of the hidden variables explanation dims if we notice that Alice and Bob are not restricted to measuring the z-component of the spin. Instead, they can measure the component along any arbitrary direction, and the result of each measurement is always either +1/2 or -1/2. Therefore, each electron must have an infinite number of hidden variables, one for each measurement that could possibly be performed.

This is ugly, but not in itself fatal. However, Bell showed that by choosing just three directions in which to perform measurements, Alice and Bob can differentiate hidden variables from quantum mechanics.