Bell Experiments

The intent of this article is to explain one of the most fascinating discoveries of physics. We’ll try to explain it as simply and clearly as possible. An advanced education is not necessary; just grade school math and the willingness to think. The discovery shows that things most of us take for granted are likely untrue. The result has been experimentally confirmed many times.

Analogy - Coloring the corners of a triangle

We’ll start with a concrete analogy here. This should help you understand the physics we explain later. Let’s say you have two crayons, red and blue, to mark the 3 corners of a triangle. You could mark all corners with the same crayon (all red or all blue) or you could mark two corners with one crayon and one with the other (2 red and 1 blue, or 1 red and 2 blue). Think about it for a minute, there’s no other option.

No matter how you mark the corners, at least one side of the triangle will have the same color at both ends. When all corners are the same color, all sides have the same color at each end; and when one corner has a different color (the only other option), then 1 of the 3 sides will have the same color at each end. We’ll call this conclusion Bell’s inequality: if someone picks a side of the triangle at random, the probability of the two endpoints having the same color is at least one third.

What quantum mechanics predicts, and experiments confirm, is that Bell’s inequality is violated! Somehow, in experiments seemingly analogous to the triangle example above, only one quarter of the time will a triangle side have the same color at each end! Keep this analogy in mind as it coincides closely with a real experiment we’ll explain now.

Experiment - Measuring electron spin

As a real, physical example of the phenomena we’d like to explain, we discuss the spin of entangled pairs of electrons. Don’t be intimidated by the terms “spin” and “entangled”—all you need to know are the facts below.

1. The spin of an electron may be measured about any chosen direction/axis (e.g. north, right, south-east, …) and the result will be either up or down about that axis.
2. If two electrons (A and B) are entangled and the spin of each is measured about the same axis, then the result (up/down) will be the same. E.g. if I measure the spin of A to be up about north, then I know a measurement of B about north will also be up. (There are other types of entanglement, but this is all we’ll use here.)

We now consider an experiment repeated thousands of times. The experiment is simply this: create an entangled pair of electrons and measure the spin of each. We’ll measure about one of three different axes A1, A2 or A3. We chose these axes to be 120 degrees separated from one another. For each electron in each trial, we’ll pick from these three axes randomly. We’ll record 4 values from each trial: the two axes we measured about and the two up/down spin results.

Per fact #2 above, we know that in all trials where we measured both electrons about the same axis, we get the same spin result. Our data will confirm this and we’ll remove those trials for now. What’s left are thousands of trials where we selected angles 120deg offset from each other.

With this remaining data, how many cases do you think had the same spin measurement (both electrons were found to be up, or both down)? You may guess half, assuming nature assigns up/down to each axis with 50% probability. This is reasonable, but let’s say mother nature desires as few spin matches as possible, with the constraints that she chooses spins for the three axes before she knows which 2 we chose to measure (and she must satisfy facts 1 and 2 above).

With some thought, you’ll see this is like the triangle coloring analogy. The best nature can do is assign a spin to one axis, and a different spin to the other two axes. In this case, only a third of the trial results will have the same spin. (To see this, think of the triangle analogy where each corner of the triangle is an axis A1, A2 or A3 and red represents spin up, blue spin down. A side of the triangle represents a choice of two axes in the experiment, and only one third of such choices will have the same color/spin at each end.)

So we have the results from all trials in which spin was measured at axes 120deg apart. If mother nature assigned spins optimally to prevent matches, she could drive the match rate down to a third. However, experiments many times over show that the match rate is just a quarter (as predicted by quantum mechanics)! So where did our logic go wrong? How could the match rate be lower than a third?

How Does Nature Violate Bell Inequalities?

Two potential explanations follow.

1. Perhaps nature knew which angles we would choose, or at least coerced us into choosing angles with different spin results many times. This amounts to a lack of free will—our choice of angle is partially or completely out of our hands.
2. Nature assigns spin results after she knows which angles we choose. In this way, she can trivially choose the same spin results a quarter of the time. This may seem like an easy explanation, but there’s trouble (see below).

Before you accept option 2, let’s expand on it a bit more. It’s not enough for nature to determine the spin of electron A based on the measurement direction at A. In order to make matches occur a quarter of the time, nature must know both of the chosen measurement axes before providing either measurement result. For her to determine the spin result for electron A, she must know how electron B is being measured (in addition to A). The trouble is that A and B could be millions of miles apart, and we could make our measurement decision at the last instant. This would require the measurement decision at A to affect B “instantaneously,” at an arbitrary distance away. Like the lack of free will discussed in option 1, this nonlocal influence is hard for some people to swallow.

It’s possible the influence is local in a more refined theory. Perhaps our understanding of spacetime is flawed. Maybe A and B are actually “right next to each other” in a more refined definition of distance. Perhaps the space we perceive between them isn’t experienced by them until after they are measured.

The main point of this article is complete. If you’re interested, there’s one more section about how quantum mechanics explains this experiment.

What does quantum mechanics say about this experiment?

According to quantum mechanics, the pair of electrons in this experiment are in a (quantum) state unfamiliar to us. This state does not foretell (before measurement) the spin result on any axis, much less all three axes! Only when measurements are made does the quantum state change to a state with definite spin about the measured axes. In the case of entanglement, both electrons, no matter how far separated, “collapse” to the same state of definite spin when either is measured. So if A is measured to be up about A1 then A and B instantly have a definite spin up state about A1. Now, if B is measured about A2 (120deg offset from A1), it has a ¼ probability of jumping from a definite up about A1 to a definite up about A2 (the probability decreases from 1 when the angle is 0, to 0 when the angle is 180deg). Unfortunately, quantum mechanics only provides these probabilities—it says nothing deeper about how they are achieved.