EPR Paradox

Section 8.5 EPR Paradox

You may have heard at some point that Einstein never accepted quantum mechanics. There is a grain of truth in it: Einstein did not dispute the predictions of quantum theory, but he never believed that quantum mechanics was a complete theory. For example, he did not accept the idea that a particle does not have a definite position. He was convinced that there must be some more information available that we simply haven't understood yet, and that if we could work out this additional aspect to the theory then particles would once again have precise positions, as in classical physics. This additional information, which is not in the quantum theory but Einstein felt must be a part of reality, is called a hidden variable.

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In trying to prove his point, Einstein, along with Podolsky and Rosen, formulated a thought experiment that leads to an apparent paradox, called the EPR paradox. Specifically, consider creating an electron-positron pair in the Bell state

\begin{matrix} (8.14) & | ψ ⟩ = \frac{1}{\sqrt{2}} | ↑↓ ⟩ - \frac{1}{\sqrt{2}} | ↓↑ ⟩ . \end{matrix}

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This is fairly easy to do: our particle accelerators create many electron-positron pairs in their collision events, and conservation of angular momentum requires that the pair have a total angular momentum of zero. Hence, they pop into existence in precisely this Bell state.

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Now imagine that you have physically separated the electron-positron pair without disturbing their spin states. Maybe, by chance, you get a pair heading directly away from each other, and you keep them from interacting with any other particles until they are a full meter away from each other.

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Figure 8.2. The EPR thought experiment. An electron-positron pair is created in state $| ψ ⟩$ given in (8.14). The particles are physically separated and then their spins are measured by Stern-Gerlach devices.

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After they are separated, you send the electron through a Stern-Gerlach device to measure its

S_{z}

spin component, as shown in Figure 8.2. The result is random, with an equal chance of spin up or spin down. But as soon as you have made that measurement, the positron spin is no longer random. Essentially, you have collapsed the state of both particles in that instant, even though they are separated by a meter.

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If that doesn't seem strange, imagine first letting the particles get a kilometer apart and then doing the measurement. Or even a light year apart. Quantum theory says the distance is irrelevant: if you can get the particles entangled and keep them entangled, then collapse of state due to a measurement “travels” from one particle to the other instantly, regardless of their separation. This is what Einstein referred to as “spooky action at a distance.” And he felt that this argument proved that quantum mechanics was not a complete theory.

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It is worth emphasizing again that Einstein did not dispute the result predicted by quantum theory. But he thought a more natural explanation was that the state

| ψ ⟩

in (8.14) is not really a complete description of the electron-positron pair. Rather, there is some more information, the hidden variable, which if understood would cause these measurements to be not mysterious at all. It helps to have a specific example of a hidden variable theory in mind to digest this.

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Example 8.3. A Hidden Variable Theory.

Imagine that, contrary to what you learned about quantum mechanical spin in Chapter 5, an electron really does have a definite spin vector $\vec{S}$ with all three components precisely determined. None of this probability stuff. So this $\vec{S}$ vector is the hidden variable.

We can reconcile this idea with the Stern-Gerlach experiment by claiming that we don't completely understand spin measurement. When we measure $S_{z},$ we somehow end up getting the value $+ ℏ / 2$ any time that $\vec{S}$ has a positive $z$ -component. And we end up measuring the value $- ℏ / 2$ any time $\vec{S}$ has a negative $z$ -component.

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This simple example of a hidden variable theory explains the EPR experimental results in a straightforward way. The electron-positron pair are created with net angular momentum of zero, so we must have

{\vec{S}}^{elec} = - {\vec{S}}^{pos} .

These

\vec{S}

vectors are established at the beginning, when the two particles are in contact. And while we don't know the value of either

{\vec{S}}^{elec}

{\vec{S}}^{pos},

we do know their

z

-components have opposite sign. So the EPR experiment will always give opposite signs for the electron and positron

S_{z}

components, but not due to any communication at a distance.

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This is a pretty compelling argument! But interestingly, the EPR paradox did not sway most physicists to abandon quantum mechanics. One reason was that Bohr showed that this instantaneous state collapse across a large distance did not actually violate special relativity. No actual object travels from the electron to the positron. But more significantly, no actual information travels either: the person measuring the electron cannot send a signal via the state collapse ( “spin up if by land, spin down if by sea” ) because they cannot control what they will find for the electron spin. They just get a sequence of random results, and the positron measurer also gets a sequence of random results. Comparison later reveals that the random results are connected: every time the electron observer measures

S_{z}^{elec}

to be spin up, the positron observer measures

S_{z}^{pos}

to be spin down and visa versa.

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The main reason that the EPR paradox did not push physicists to abandon quantum mechanics, though, is that it didn't seem to matter. After all, the hidden variable theories predicted the same results as quantum mechanics. Since physicists knew how to use quantum mechanics and it kept giving successful predictions about the microscopic world, there was little motivation to worry about whether we should replace quantum mechanics with a hidden variable theory.