Documentary: The EPR Paradox & Bell's inequality explained simply

Introduction

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Two of the biggest intellectual giants of modern physics changed our view of reality,

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and proposed their theories around the same time, in the early twentieth century.  Yet,

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their views about reality were completely different.

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Niels Bohr, one of the founders of modern quantum mechanics and the father of the Copenhagen

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interpretation, argued that reality, or the state of a particle at the fundamental

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level, was not only unknown, but was unknowable, until it was measured.

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Albert Einstein passionately disagreed with this idea, and believed that reality was knowable

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and that probabilities could not completely define reality, that there had to be an objective

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reality out there, independent of measurement.  He famously said, in his dramatic statement,

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“Do you really believe the moon only exists when you look at it”

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Bohr and Einstein argued passionately about their views on the essence of reality. And

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for 30 years, both views were considered equally valid, and scientists chose sides.

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Then in 1964, Irish physicist John Bell devised a way to prove whether Einstein’s view of

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a classical, deterministic view of reality was correct, and he put this in a simple elegant

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equation – called the Bell Inequality.  Although the equation is simple, understanding

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what it means is not so easy. So how did this simple equation disprove the ideas of the

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most famous and powerful scientist of the 20th century, and forced us to confront the

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unsettling truth that we may live in a fundamentally nondeterministic universe.  The answer is

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coming up right now...

The Copenhagen Interpretation

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The central weirdness of quantum mechanics

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can be demonstrated with a toss of the dice. If the dice was a quantum system like a photon, electron

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or atom, it would be in superposition. That is, its position and other properties, in

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a sense would be up in the air like a dice thrown before it hits the table. It is a 1,

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2, 3, 4, 5,  and 6 all at the same time. According to the most accepted interpretation

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of quantum mechanics, the Copenhagen interpretation, pioneered by Neils Bohr and Werner Heisenberg,

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its not only that the dice’s value is not known, but that it is all values at once.

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It’s value can only be known once it is measured.

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The double slit experiment demonstrates this.

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A single photon passing through the slit should not be thought of as a particle going through,

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and interacting with itself. Rather, it is like a three dimensional wave or a cloud of

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probability when it is emitted. This wave can go through both slits at once and interfere

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with itself like a wave does, before it is measured by the photo sensitive screen in

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the back, when it resolves to a specific location on the screen.

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Einstein, who was the most famous scientist

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at the time, was bothered by this interpretation of quantum mechanics. The idea that part

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of nature remains not only unknown, but is unknowable unless measured. That’s why he

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famously said, “God does not play dice.”

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So Einstein along with two other scientists, Boris Podolsky and Nathan Rosen (EPR for short),

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came up with what they thought disproved the Copenhagen interpretation. The crux of their

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argument rested on the idea of a phenomenon predicted in quantum mechanics called entanglement.

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Let’s say we have a quantum system that

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has a zero angular momentum, also known as spin zero, and it emits two photons simultaneously.

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Since photons have spin, and since angular momentum must be conserved, if one photon

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has a spin up, the other photon must have a spin down. So the spin up and down cancel

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each other out. This is entanglement. The two photons are not independent. If the spin of

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one of them is known, the spin of the other becomes known instantaneously.

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Now let’s separate these photons far apart,

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let’s say 300,000 km, which is about 1 light second away. Now according to quantum mechanics,

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each of these photons is governed by a wave function, and each photon is in a state of

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both spin up and down at the same time. It is not up or down, it is both up AND down.  However,

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the wave functions of these photons is not independent. They are really the same wave

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function governing both photons. Now if we measure the spin of one of them. We have a

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a 50% chance of detecting a spin up. Let’s say do we detect spin up.  Now let’s

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measure the spin of the other photon in less than one second, it should also have a 50%

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chance of having spin up or spin down. But it doesn’t, it has a 100% chance of having a spin down.

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This means that the information of the collapse

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of the wave function of one of the photons had to travel faster than light to affect

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the other photon. EPR argued that since nothing can travel faster than light according to

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the rules of special relativity, this should invalidate the Copenhagen interpretation. This theorized

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violation is the EPR Paradox. EPR proposed instead there is likely another

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theory which would show that the two entangled photons were in cahoots from the very beginning,

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that their states were predetermined from their creation. In other words, their states

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contained the information locally, so that when they were moved apart, no communication

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had to take place. The information that we were measuring was hidden inside the two particles.

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This is called local hidden variables.

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So for example, it’s as if the two particles were a pair of gloves. One was a left handed

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pair, and the other was a right handed pair. Once we found the left handed pair, we knew

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immediately that the other pair, no matter where it was in the universe, must be a right

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handed pair. This was a valid interpretation of quantum mechanics for almost 30 years,

Bells Inequality

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From 1935 to 1964. This could not be disproven.

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This is where Irish physicist, John Bell comes in. And in 1964, he proposed an experiment that

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could show whether the local hidden variable theory was correct, or incorrect.

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Bell’s equation is remarkably simple. But,

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it is one of the most difficult to understand and simplify. So I will present a highly simplified

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idea of how it works, then also show you simplified version of the mathematical underpinnings

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later in this video.

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We are going to play a game, and our opponent will be the universe. The game is guess the color of the checkers pieces. I want

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to give a big shout out to Jim Al-Khalili who shows a similar simplified illustration

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in a documentary available on Magellan TV. I encourage you to check it out because it

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is fascinating.

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If I guess the color correctly I win, if I am incorrect, the universe wins. In the first

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game I declare that if the colors match I win. The universe picks the pieces. In play

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after play, I find that I lose every time.  So in the next round, I change the rules.

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This time, I declare that if the colors are different I win. The universe picks the pieces.

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In play after play, I find that I lose every time again.

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So, I conclude that perhaps the universe has rigged

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the game against me. This is what Einstein suspected was happening, that the checkers

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Piexes were rigged. He believed that the colors of the pieces were predetermined, and that

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this information had been known from the beginning when the two pieces were together. Niels Bohr’s

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idea, on the other hand, was that the red and black don’t even exist until the piece is turned over.

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So, I change the rules of the game again.

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This time, I do not tell the universe whether matching colors, or different colors will

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win the game, until after the universe has already picked the checker pieces.

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If after revealing the checker pieces, I find that I win 50% of the time, and lose 50% of

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the time, then my conclusion is that indeed the checker pieces were rigged from the very

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beginning, because my chances of winning would be 50% if the red and black pieces were already

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picked by the universe. But if I continue to lose, then I have to conclude that somehow,

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the colors of the checker pieces were not chosen ahead of time.

Bells Elegant Equation

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Let’s look at how Bell’s elegant equation proves my conclusion. To do this, let’s

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look at a universe where local hidden variables are correct.

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In other words, entangled particles have predetermined

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properties which are complementary to each other at the moment of their creation. Note

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that this is not an interpretation of quantum mechanics. It’s an alternative to quantum

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mechanics.

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So in this system, when two entangled particles are produced from a process that conserves

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linear and angular momentum, we will get two particles that go off in opposite directions,

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and they will have opposite spins, as well as opposite directions.

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Let’s say that Alice makes the measurement

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on particle 1 in her laboratory, and Bob makes the measurement on particle 2 in his laboratory.

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And let’s say that the two labs are very far apart and are not in communication with

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each other.

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If Alice measures the particle’s spin in the Z direction as positive, Bob will

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measure the spin as negative in the Z direction. If they measure the spin in the X direction

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which is orthogonal to the Z direction, or 90 degrees apart, the same type of complimentary

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opposite spins will be measured, because, as I said earlier, angular momentum is always

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conserved.

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What Bell’s inequality does is, it says, now let’s make a third measurement that

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is somewhere in between the Z and X axes. In our example we will use 45

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degree angles to the Z and X direction, And we will call this the Q direction.

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So in a universe where local hidden variables

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are true, when the two particles are emitted, they know what their state is going to be

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in all three directions, Z, X, and Q from birth. And there are only 8 combinations or

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possibilities of spins that each particle could have. So for Alice and Bob, these 8

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combinations will be following:

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Event 1 could be where the spin in the Z direction is positive, the spin in the X direction is positive, and the spin in the Q direction is posive. positive

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Event 2: Z is positive, X is positive, Q is negative

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Event 3: Z is positive, X is negative, Q is positive

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Event 4: Z is positive, X is negative, Q is negative

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Event 5: Z is negative, X is positive, Q is positive

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Event 6: Z is negative, X is positive, Q is negative

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Event 7: Z is negative, X is negative, Q is positive

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Event 8: Z is negative, X is negative, Q is negative

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Now, let’s ask the question, what is the

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probability that Alice measures in the z direction, gets a positive spin, and Bob measures in

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the X direction and gets a positive spin? Well, if the above case is for Alice, there

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are 4 events where Z is positive. In order for Bob to get X positive, Alice would have

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to have measured X as negative. So these would be in event 3 and event 4. To get the probability

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we have to divide by the total number of events, which is 8.

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Let’s do this for two more scenarios. What is the probability that Alice measures positive

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in the Z direction, and Bob measures positive in the Q direction? In this scenario, it would

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be event 2 and 4. Again we divide by 8 to get the probability.

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And the third case is: What is the probability that Alice measures positive in the Q direction,

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and Bob measures positive in the X direction? This would be event 3 and event 7, divided

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by 8 for probability.  So these are the three probabilities given

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the hidden variables theory. Now here is the big insight that John Bell had:

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If I take the total number of Events, and multiply that by the probability that Alice

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measures Z positive and Bob measures X positive, this has to be less than or equal to the total

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number of events times the probability that Alice measures Z positive, and Bob measures

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Q positive, plus the probability that Alice measures Q positive, and Bob measures X positive.

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When we write this out,

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the eight cancels out, and we are left with just the probabilities:

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This is Bell’s inequality.

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I can prove this is true by doing simple math: E3 + E4 divided by 8 is less than or equal

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to E2 + E4 + E3 + E7 divided by 8. The eights cancel out, and we rearrange the

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order of addition, and we are left with this: E3 + E4 is less than or equal to E3 + E4 +

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E2 + E7 This makes total sense, because E3 and E4

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are on both sides of the equation. And E2 and E7 have to be positive. So this inequality

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absolutely HAS to be true for any hidden variables theory to be true. Now remember these probabilities

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are for a universe with hidden variables.

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But what happens in a universe where the laws of quantum mechanics are correct, and not

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the hidden variables theory?  Well, this inequality is violated in quantum mechanics. How is it

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violated?  Let’s say Alice measures the spin to be positive in the Z direction. Then,

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we know that if Bob measured the particle in the Z direction, he would get a spin that is negative.

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However, Bob doesn’t measure in the Z direction,

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but in the Q direction. What will the spin of his particle be?  In hidden variables,

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there was a 50% chance that it would be positive, and a 50% chance that it was negative. But

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this is not what happens in quantum mechanics, because the measurement of the particle follows

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the probability laws of the wave function for a particle rotated 45 degrees.

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And that probability of Bob measuring Q to

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be positive, after Alice has measured Z to be positive, if the angle between them is

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45 degrees, is given by the following equation:  sine sqared of 45 degrees divided by 2.

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This comes from the math of quantum mechanics. This is the critical difference between quantum

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mechanics and hidden variables theory. The probability that Bob measures the same spin

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as Alice, depending on the difference in angle measured is not linear but looks like a sine

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wave. When you plot this out, this is what the probabilities look like:

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So you can see from the graph that at 0, and

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multiples of 90 degrees, the two systems are in agreement. But in between, like at 45 degrees,

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the probability is 25% for hidden variables, and about 14.6% for quantum mechanics.

Bells Inequality is violated

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But the proof is in the pudding, because in

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test after test, the sine function correlation has been confirmed. The particle does not behave

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linearly. And so the hidden variables theory cannot be correct. Bell's inequality is violated.

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So we can write out and compare Bell’s inequality

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for both cases.

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For hidden variables, the equation would be: E3 + E4 is less than or equal to E3 + E4 +

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E2 + E7 2 is less than or equal to 4 – the inequality

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holds true.

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For quantum mechanics, the equation would be:

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Sine squared of 90 degrees divided by 2 is less than or equal to sine squared 45 degrees

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divided by two plus the sine squared of 45 degrees divided by two.

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And when you solve this, you get: 0.5 is less than or equal to 0.293,

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which, of course, is not true, so Bell’s inequality is violated in quantum mechanics, which is

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exactly what we observe in experiment after experiment.

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Does this prove that the two entangled particles

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are communicating faster than light? There is differing

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opinion on this. On the surface, it does appear that superluminal communications is taking

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place. But my personal opinion is that no communication is happening. The two particles

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are really just part of one wave function, and since the wave function can theoretically

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be as wide as the universe, when this one wave function collapses, both particles collapse.

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And since the collapse is random, it can not be used to communication in any way.

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So most theorist do not think special relativity

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is violated, because we can’t communicate using this seemingly faster than light phenomenon.

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But our knowledge of quantum mechanics and

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entanglement is incomplete to say the least. Why is there a superposition? What is the mechanism of wave

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collapse between two particles separated by space and time? I happen to think that the

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scientist who is going to reveal the mystery of quantum mechanics is alive right now in

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the world. And I can’t wait for him or her to explain this, and like Copernicus, Newton

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and Einstein -  completely change our paradigm. I’ll see you in the next video my friend.