The Bell Inequalities

Albert Einstein was so smart that even some of his mistakes led to entire fields of research.

In 1935 Albert Einstein published a paper with Boris Podolsky and Nathan Rosen[6] outlining what is now called the EPR paradox, arguing that quantum mechanics must be incomplete based on the following thought experiment: start with two particles which can be measured to have physical properties of position x and momentum p, i.e.

Start with two states ψ1(x1, p1) ≡ |ψ1〉 and ψ2(x2, p2) ≡ |ψ2

The EPR experiment then follows:

1. Interact the particles for a time such that they become maximally entangled

   We cannot describe them as two independent states |ψ1〉 and |ψ2〉, but rather as a single two-particle composite system. They have equal and opposite values of momentum p1 = −p2 when measured in momentum space, or corresponding values x1 = x2 + x0 for some constant x0 when measured in position space.

2. Separate the particles in opposite directions by a large distance, let’s say a light-year apart.

   1. Because of the nature of entanglement, measuring p1 for |ψ1〉1 with certainty (i.e. ∆p  ℏ · [L−1]) simultaneously gives you the information to know p2 with certainty. Likewise, measuring x2 for |ψ2〉 with certainty simultaneously gives you information to know x1 with certainty.