Bell's Theorem - Explanation, Bell's Inequality, Local Realism

Introduction

Bell’s theorem is a very interesting concept of modern science and philosophy. The strange realm of quantum mechanics is often complicated, and at times, incalculable. More importantly, investigating quantum mechanics often leads us to the interface between the world of science and philosophy. You might not know this, but these two topics are closely related. It might interest you to know that some of the most prominent paradoxes in Physics were proposed by philosophers, not physicists or scientists. Achilles and the Tortoise is a famous example of the latter.

Another concept that, in a way, bridges the gap between philosophy and science is what is commonly known as Bell’s theorem. The theorem was proposed by John Stewart Bell in 1964. There are various ways to state this theorem but the simplest sentence I could find was the following:

“No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.”

Interestingly, you should understand that Bell’s theorem only talks about a theory of local hidden variables. That is, perhaps hidden variables are there in this world. What Bell’s theorem says is that a theory that works with local hidden variables cannot exist side by side with what quantum mechanics has predicted. In this article, we are going to discuss Bell’s theorem in detail.

What does Bell Theorem mean?

Quantum mechanics says that the act of measurement forces the system to take a stand. For example, before you measure the position of a subatomic particle, its position could be anything. It might not even exist! However, when you perform the “measurement”, the particle in question “acquires” the property you are measuring. Notice how “measurement” is in quotes here. I do that because as such, you cannot “measure” the position of an electron using a ruler or GPS tracking. Measurement in this scenario could mean calculation, observation, or any other act that gives you the answer you seek.

Now, at this stage, you might be thinking how could that possibly make sense? How can a particle magically “acquire” information about the property you are measuring due to the act of measurement? As it turns out, Einstein also had his doubts about this idea and in a paper in 1935, along with Boris Podolsky and Nathan Rosen, he proposed that there is no magical acquisition of property. Instead, there are “Hidden Variables” in the system. To quote Einstein,

“I like to think that the moon is there even if I am not looking at it.”

This is where Bell’s theorem comes in. His theorem shows that what Einstein proposed cannot always be. If there were local hidden variables, then in certain cases, we would arrive at stages where we would disagree with the established predictions of quantum physics.

Bell’s Theorem Formula

Mathematically also, there can be multiple ways to state Bell’s theorem. However, in the most rudimentary of senses, here’s what we can say −

$$\mathrm{P(X=Y)+P(Y=Z)+P(Z=X)≥1}$$

Here, X,Y,and,Z are random variables that can take the value of ±1. That is, they are the possible outcomes of three measurements that do not affect each other.

What is Bell’s Inequality?

Mathematically, Bell’s theorem is an inequality that we refer to as Bell’s inequality.

$$\mathrm{P(X=Y)+P(Y=Z)+P(Z=X)≥1}$$

Now, remember that Bell proved this inequality to be true. However, if you use logical calculations, or try to find the probabilities of events you observe in the real world, you will arrive at situations where this inequality would not be satisfied.

Interestingly enough, no one knows why this happens. There is no exact answer to why Bell’s inequality is not always satisfied. However, Bell’s theorem itself makes one thing clear: If there were local hidden variables, they wouldn’t be able to satisfy all the predictions of quantum mechanics. Perhaps some, but never all.

What is Local Realism?

Local Realism is a principle satisfied by classical mechanics which encompasses two principles −

Locality

Locality means that for some sort of physical change to occur, some sort of physical action has to take place. For instance, if you wish to make an almirah, you have to physically nail its walls and doors in place. In a local universe, you cannot bring about physical change without bringing about physical action, i.e., touch.

Realism

When we talk about realism, we mean to say that the universe in question “really”, objectively exists outside of our minds. Regardless of whether we are looking at it, touching it, or changing it, the universe is there.

Now, compare this principle to what we discussed about quantum mechanics in the beginning. We said that “the act of measurement forces the system to take a stand”. Thus, quantum mechanics operates outside the principles of Local Realism.

Conclusion

In the strange world of quantum mechanics, science and philosophy often get tangled up. Bell’s theorem is a remarkable example of this tendency. To understand Bell’s theorem, it is necessary to understand how quantum mechanics works.

According to quantum principles, the properties of a system do not exist outside the context of observation. Instead, when we perform a measurement, we force the system to acquire or assume a certain value for the property which we are measuring. In a paper, Einstein, Boris Podolsky, and Nathan Rosen proposed that this assumption is unreasonable and that we must have some sort of hidden variables in the system. John Stewart Bell, in 1964, extrapolated on this idea and arrived at a theorem that refuted the necessity of hidden variables. In its simplest form, it states that a theory that believes in local hidden variables will fail to satisfy all the requirements of quantum physics. That is, in some cases, the presence of local hidden variables will contradict what quantum mechanics predicts.

The concept of local realism is closely related to the study of Bell’s theorem. Local Realism is a principle satisfied by classical, but not quantum mechanics. It encompasses two ideas, viz. Locality and Realism. That is, physical change requires physical touch or that information cannot travel faster than light in free space, and that the universe objectively exists outside of our minds.

FAQs

Q1. Why is Bell’s theorem significant?

Ans. Bell’s theorem proves that quantum mechanics defies the principles of local realism and thus, is an important proof in favour of the views of quantum mechanics.

Q2. What do you mean by local variables?

Ans. Locality means that the variable is not instantaneously affected by events that are far away. Information of any sort of change or “cause” travels at the speed of light and the “effect” of this change is visible only after the delay involved in this propagation.

Q3. Does Bell’s theorem invalidate quantum mechanics?

Ans. No. Bells’ theory simply says that a theory that believes in local hidden variables cannot satisfy the predictions of quantum mechanics. Thus, either hidden variables exist, or what quantum mechanics has predicted for us is correct. But the two cannot be simultaneously true.

Q4. Do hidden variables exist?

Ans. No. It is the current theory of quantum mechanics that has been proven to be correct. The existence of local variables has been refuted by various experiments.

Q5. Give an example of an experiment that can be used to analyze Bell’s theorem.

Ans. If we measure the possible polarization states of a photon, we can analyze the predictions of Bell’s theorem.