Bell's Theorem and Implications

Hidden Variables

When learning about quantum mechanics, one of my first instincts was to think:

We “can’t predict” the spin of an electron? It’s “random”? Well, it’s probably just because we aren’t smart enough yet, or we just don’t understand it, or we don’t have good enough equipment.

In the past, we thought the weather was “random” and couldn’t predict it. In the future, we’ll say the same thing about quantum mechanics, and why an electron spins up or down.

The idea that wave functions are controlled by some sort of secret process is called a hidden variable theory. If you also assume that the speed of light is the speed limit of the universe, your theory is technically a local hidden variable theory. It’s a popular idea— even Einstein was a fan at one point¹.

Even if we can’t measure local hidden variables, is there any way to tell if they exist?

Yes! In 1964, John Bell came up with Bell’s Theorem to help answer this question. Bell came up with equations, called Bell inequalities, that must be true if local hidden variables really exist. Once such inequality is the CHSH inequality:

What does this equation mean? What are and ? To understand how Bell arrived at these inequalities, let’s discuss a kind of experiment called a Bell test.

The Experiment

1. Alice and Bob entangle 2 electrons.
   • As a reminder, this means the electrons have a special quantum relationship.
   • When we later measure the “spin” of each electron, it will either be “spin up” (represented as ) or “spin down” (represented as ).²
2. Alice and Bob each take one of the entangled electrons, and a coin (which they will later flip).
   • We’ll call Alice’s coin flip , and Bob’s coin flip :

3. Alice and Bob come up with methods of measuring the electron’s spin, according to their coin flip. Their methods are defined by the angle by which the electron’s spin is measured.² Let’s say they come up with these methods:

We can define 4 variables based on the observed spins of the electrons:

Within a single trial, only 2 of these options will be measured: Alice measures either or , and Bob measures either or . Since their results are based on their coin flips, we call their results and .

4. Alice and Bob travel very far apart and perform the experiment, writing down the result. They repeat this many times. For example, 10 rounds may look like this:

CHSH Quantity ()

In any single experiment, only one pair of outcomes will be generated: . We define the correlation by multiplying the resultant spin’s values (see column above).

We can now calculate the average correlation, , of any pair by averaging all the correlations in experiments that involved :

For example, to calculate , we look at the experiments in which occurred (Alice flips tails, and Bob flips heads). In our scenario, experiments 3, 6, and 10 fit that description. Now, we average the outcomes of those experiments:

Using these values, we define a special value called the CHSH quantity:

After the experiment has been done many, many times, we calculate the CHSH quantity .

Classical Conditions

Now, let’s imagine that there is a local, hidden variable that tells each electron how to spin when measured a specific way. Imagine each electron has these secret instructions:

• for Alice’s 0°
• for Alice’s 90°
• for Bob’s 45°
• for Bob’s 135°

Since and , that means either or . Thus, it should be intuitive that:

Since either or will be zero, only the other term will be left, and that remaining term will be .

Now, we can distribute the terms and move forward:

We again define based on average correlations.

Using , we re-write the above equation:

Now, we see the CHSH inequality that is met in any local hidden variable scenario.

Quantum Conditions (Real World)

Remember that in a classical scenario, the CHSH inequality is always met. When we do these experiments in the real world, however, we observe that the correlation is based on the relationship between and :

Now, we can calculate for each of our possible outcomes:

Plugging these equations into the CHSH inequality, we get:

Thus, we have:

This is a violation of the CHSH inequality!

Implications

If there were local hidden variables secretly controlling how an entangled particle should spin, we would observe . But in the real world, we observe !³

Because of these experiments, we know that local hidden variables are not enough to explain the observed phenomena of quantum systems. We’re left with these options:

1. There are still hidden variables, they’re just not local. They’re far away, and must be interacting with the particles at faster-than-light speed.
   • This violates locality, or the idea that no information can travel faster than light.
2. There are no hidden variables; the measured spin is just probabilistic. Particles don’t have a definite spin before you measure it.
   • This violates realism, or the idea that properties (e.g. the spin of an electron) really exist before you measure them.

All valid quantum interpretations must either violate locality, or violate realism (or otherwise escape this restriction, as clever physicists sometimes do). For example:

• The Copenhagen interpretation violates realism: it says quantum systems don’t have definite properties before measurement.
• The Pilot Wave interpretation violates locality: it says there is a universal “pilot wave” that always knows the position of all particles in the universe, thus requiring information traveling faster than light.

In the next article, we’ll discuss these two interpretations in more detail, and a few more.