Explaining the EPR Paradox

For anyone in their first year of quantum mechanics, the Einstein-Podolsky-Rosen Paradox (EPR) is required study.  Its significance is underlined because it marked the final confrontation between the classical way of viewing the world, and the emerging quantum way.  At its heart, is the universe deterministic, predictable, and classical?  Or is it random, indeterministic, and quantum?  Models describing particles with wave functions had emerged and been tested, and there was no question QM worked (and worked well).  The question was, is QM fundamental?  Or are all these probabilities and uncertainties the result of our particular model, which covered up a misunderstanding of deeper principals with probability descriptions?

Einstein, father of the beautifully deterministic worldview given to us in general relativity, refused to believe the universe could operate, at its lowest levels, with uncertainty.  This is what he meant when he famously said, “God does not play dice with the universe”.  It is also misquoted rather vigorously to suggest Einstein meant a literal god, but the fact he often spoke pantheistically is well documented.  In any event, he felt strongly that a deeper understanding of quantum principals might some day remove the uncertainty and show us what was really going on under the surface.

Along with Podolsky and Rosen, he proposed a paradox which highlighted some concerns he had in the early formulations of quantum theory.  Without going into the specifics, which Wikipedia can cover better than I, here is the important assertion raised by the paradox:

If we produce two particles in a twin state (entangled) are fired them off towards detectors, their spin is described by QM as being in a superposition of states, and is thus effectively undefined until measured.  EPR suggested the particle might be random in-so-much as it was assigned randomly at the time of entanglement, but while in flight and prior to being detected both particles must have (the same) determined (x,y,z) spin.  It was not, fundamentally, uncertain.

Nobody could test this, so it was left to ponder, that is until Bell came along with an experiment (untestable at the time) which could set the record straight, known as the Bell Inequality.  Here is what Bell figured out:

Produce your entangled particles, then setup two detectors, one to measure each particle.  The detectors are setup to RANDOMLY measure only 1 axis each time, either X, Y, or Z.

If a particle has a determined spin, it can be described as (A, B, C), where A, B, and C are either U (up-spin) or D (down-spin) for the x, y, and z axis respectively.  (U,U,D) would be a particle spinning up on the x and y axis, and down on the z axis.

So in Bell’s setup, each time we run the experiment, each detector would report either “UP” or “DOWN” for whatever it measured on the axis it randomly chose that time.  The two detectors pick their axis at random and need not report which they used, thus each time we run this experiment there are 9 combinations of measurements we might see.

Detector 1 Detector 2
x x
x y
x z
y x
y y
y z
z x
z y
z z

In each of those 9 cases, we will just see UP or DOWN from each detector, not knowing which of those 9 specific combinations led to that result.

Quantum mechanics tells us that when the particle is measured, no matter what axis we choose, the result will be U 50% of the time, and D 50% of the time.  Therefore, half the time the two detectors will agree, half the time they will not, as you can see below:

D1: U, D2: U => Agree

D1: U, D2: D => Disagree

D1: D, D2: U => Disagree

D1: D, D2: D => Agree

The quantum case is easy.  If EPR were wrong, and we run Bell’s experiment, we should see the two detectors agree 50% of the time.

The classical case is a little longer, though not complicated.  Here is the point that will become the key: if we have 3-axis spin, such as (U,D,U), at least two of the axes will ALWAYS be the same.  Think about it.  You can only assign U or D to each axis, and you have 3 to fill, so no matter how you do it, at least 2 will match.  Keep this in mind because it is the foundation of Bell’s inequality and breakthrough.  Here are the 8 possible combinations for particle states, classically speaking:

(U,U,U), (D,D,D), (U,U,D), (U,D,U), (U,D,D), (D,U,U), (D,U,D), (D,D,U)

Let’s dig into this a bit.  The first two I listed above will cause our detectors to ALWAYS agree, no matter what axis they pick.  If the particle is in state (U,U,U), then it doesn’t matter if I look at x, y, or z… I am going to see a U, and so will the other detector (remember, for our simplified example entangled particles have the same state).  This means out of the 8 possible particle configurations, we can expect 100% agreement 2 times out of 8 (no matter which of the 9 axis-combinations chosen by our detectors).

The other 6 cases (U,U,D), (U,D,U), (U,D,D), (D,U,U), (D,U,D), (D,D,U) will always agree 5/9th of the time.  How did I figure this out?  We can see it visually.  Every particle in these 6 cases as a majority spin and a minority spin, represented by the solid and open dots respectively in the below diagram (the three on the left are the particle hitting detector 1, and the three on the right are the particle hitting detector 2).  I’ve drawn in the 5 combinations where the two detectors will agree, e.g., if detector 1 reads the first majority axis and detector 2 reads the first majority axis, they will agree, and I’ve drawn the top-most horizontal arrow to show this:


Out of the 9 possible combinations of detector-axis-choice shown in the table further up, the 5 shown right here will match, and the 4 I didn’t draw will not.

So in total, we have 2 cases out of 8 where the detectors will agree 100% of the time, and 6 cases out of 8 where the detectors will agree 5/9ths of the time, or:

Screen Shot 2016-03-13 at 14.30.58

And there we have it.  If QM is right, our detectors will spit out matching reports of UP and DOWN exactly 50% of the time.  If EPR is right, our detectors will spit out matching reports of UP and DOWN 66% of the time.

Later, the experiment was conducted, and the results agreed spectacularly with the Quantum Expectation.  This means particles really don’t have hidden properties as they flit around, they exist in a true state of uncertainty.