Much ink has been spilled on trying to understand and explain quantum mechanics. Particle-wave duality, Heisenberg uncertainty principle, void fluctuations: All these phenomena fly in the face of our intuition.
Among these phenomena, Bell’s theorem reveals one of the strangest properties of Quantum Mechanics. In essence, it shows that classical ideas cannot reproduce quantum mechanics, unless we’re willing to accept other consequences such as faster-than-light propagation. The original argument from Bell is somewhat intricate, so here we will present an easier-to-follow line of reasoning outlined by David Mermin and Sydney Coleman.
Consider the following setup that involves you and two of your friends. All of you are equipped with a machine that can measure two quantities, say $A$ or $B$. Each outcome takes the value +1 or -1. I’m going to send to each of you the same signal at regular interval. You don’t know how the signal I’m sending you is generated.
At every trial you’re free to choose which quantity $A$ or $B$ you want to measure. Now to make sure there is no coordination between the three of you, you’re all sitting a different places on Earth far away from each other such that you cannot communicate which measurements you’re about to do, even by transmitting signals at the speed of light (in technical terms, you’re in different causal regions of space-time). In other words, if you agree that signal cannot be transmitted faster than the speed of light, the outcome of your measurement is independent of what the others are choosing to measure.
As we start the experiment, you all record which variable (A or B) you measured and the outcome (+1 or -1), generating a series like this $$ A_1 = +1, A_2 = +1, B_3 = -1 $$ $$ A_1 = +1, A_2 = -1, B_3 = -1 $$ $$ B_1 = +1, B_2 = +1, A_3 = +1 $$ $$ … $$
After millions of such measurements, you come back together and compare notes. It quickly appears that every time someone choose to measure $A$ and the others $B$, the product $$ A_1 B_2 B_3 =1 $$ and similarly for the other permutations: $B_1 A_2 B_3 =1$ and $B_1 B_2 A_3 =1$.
From basic algebra you’d conclude that $$ A_1 A_2 A_3 =+1. $$ Sometimes two $A$s take the value $-1$ or $+1$, but whatever signal I’m sending the product of the three $A$s equals 1.
You now turn out to validate your conclusion by looking at the data. Behold: the data shows the opposite result: $$ A_1 A_2 A_3 =-1 $$ for all relevant trials! (and similarly for the other permutations)
What is happening here? The signal was generated by measuring the properties of entangled quantum particles. Entanglement happens when a group of particles are created or interact in such way that what happens to one particle determines what happens to the other, even if they are really too far apart to affect each other.
This creates a behavior that does not follow any classical expectations (for the technical readers, the signal was generated by measuring $A=\sigma_x$ and $B=\sigma_B$ of a three spin one-half particles state $[\ket{+++}-\ket{- - -}]/\sqrt{2}$).
We have thus shown that there are quantum mechanics properties that cannot be reproduced by classical mechanics (or even classical logics).
So we face a dilemma:
Either quantum mechanics doesn’t follow our intuition of classical ideas, or we must accept that faster than light communication between devices is possible. It’s one of the other.
This is what Bell showed - in a different form - in his original derivation, and which was later confirmed experimentally using photons. The experimental verification of Bell’s predictions was awarded the Nobel Prize in Physics in 2022.
This quantum behavior of course is counterintuitive. That said, as Sidney Coleman pointed out, much of the confusion around quantum mechanics comes from the fact that we approach it backward: We’re trying to explain a fundamental law (quantum mechanics) in terms of a less fundamental one (classical mechanics). No surprises it doesn’t work.
“Every successful physical theory swallows its predecessor alive. But it does so by interpreting the concepts of the old theory in terms of the new, NOT the other way around." Sydney Coleman
The right questions are then: How can we derive classical mechanics from quantum mechanics? And what are the implications of this quantum behavior?
Further reading
- D.N. Mermin, Quantum mysteries revisited. Am. J. Phys. 58, 731 (1990).
- Sidney Coleman, Quantum mechanics in your face
- The Nobel Prize in Physics 2022
