1.7 EPR with spins
Relative to the spin filter, we define a measurement as “spin-up” when an electron passes through the filter with its spin pointing in the same direction as the filter’s arrow.
In his paper, Bell uses Bohm’s idea of replacing measurements of position and momentum, which would be difficult to obtain, with the relatively easy detection of the direction of a quantum spin (spin–up or spin-down) using a quantum spin filter. Bohm envisaged a source emitting two entangled particles (say, electrons) travelling in opposite directions, as in the 1935 EPR paper, but with the difference that the quantum wave function selected to describe the pair of entangled electrons is what is called a singlet state.
Now, it is a feature of the singlet state that the total spin of the two particles is always zero. This is equivalent to saying that the total momentum of the two particles in the EPR paper is always zero, and, continuing the analogy, if Alice measures the spin of her particle and finds it spin-up, then she knows that, if Bob’s spin-filter is in the same orientation as her own, he will find that his particle is spin-down, meaning that the total spin is zero, as required by the singlet state. Knowing this, Bob turns his spin-filter upside-down, so that, relative to the orientation of his spin-filter, he will now say that his particle, like Alice’s, is also spin-up. In that case, Alice and Bob agree that the outcomes of their measurements match each other (both spin-up). Equally, if Alice finds that the spin of her particle is spin-down, then Bob will find, with his spin-filter orientated oppositely from Alice’s, that his particle is also spin-down, again, matching Alice’s result.
Measurements of the spins of two electrons in the singlet state always match if the pair of detectors are parallel-opposed, regardless of the orientation of the pair.
Since the electron spin will always be measured as either just spin-up or spin-down, regardless of the direction in which you measure the spin, we can incline the spin-filters at any angle and, so long as the spin-filters are parallel and opposite in direction to each other, we shall always find that Alice’s result matches Bob’s.
We may imagine Alice and Bob arranging to orient their spin-filters at a mutually agreed angle, still parallel and opposed, at the last moment before either electron is detected, and we shall still find that the outcomes match.
