Quantum Mechanics In Your Face

I was looking up David Deutsch when I stumbled across this amazing lecture by the gloriously dorky Sidney Coleman.  It's a grainy recording of a talk he gave to the American Physical Society in 1994.  It's full of witty repartee as he talks about his philosophical struggles with QM prior to embracing Many Worlds.  He wonders, for example, who or what exactly can collapse a wavefunction.  The Copenhagen Interpretation doesn't - cannot - answer this.  Coleman describes going to a colleague and sharing his concern that maybe solipsism is the only consistent interpretation - that maybe he is the only one capable of collapsing wavefunctions.  The colleague puffs on his pipe and says: tell me, before you were born, was your father able to collapse wavefunctions?  Wonderful! Quantum Physics meets Psychoanalysis.

After watching the video I made some notes on my favorite app.

I'd wanted to understand Bell's Inequality/Bell's Theorem for some time, but I'd never found the resource that could explain it using just the tools available to me.  But by accident, I'd stumbled into exactly what I had been looking for!  To understand Coleman's lecture, or my notes, you probably need 1st year undergraduate QM, specifically you will need to understand Hermitian Operators and bra-ket notation.  However, I'll attempt to explain it now without any pre-requisites.

What follows is not exactly the same as what Bell came up with in 1964.  Instead it is a logically equivalent, but - as Coleman says - pedagogically superior thought experiment invented by Greenberger, Horne, and Zeilinger. So, here goes:

Imagine an experiment in which Alice sends 3 objects: one goes to Bob1, one goes to Bob2, and one goes to Bob3.  Each Bob can do one of two experiments on their object: read value A, or read value B, and the values are always either  +1 or -1.  But the experiments are destructive - i.e. reading A changes the value you get if you then read B, and vice versa.

Now imagine the following scenario: the experiment is repeated thousands of times and each time, each Bob chooses at random whether to read A or B.  But whenever an A and 2 Bs are chosen the product is always +1. So, e.g.

• $A_1 = +1, B_2 = -1, B_3 = -1$ is possible
• $A_1 = -1, B_2 = -1, B_3 = +1$ is possible
• $A_1 = +1, B_2 = -1, B_3 = +1$ is not possible

The final observation is that every time 3 As are chosen the product is always -1. So

• $A_1 = -1, A_2 = -1, A_3 = -1$ is possible
• $A_1 = -1, A_2 = +1, A_3 = +1$ is possible
• $A_1 = -1, A_2 = -1, A_3 = +1$ is not possible

So what does this tell us?  I think to understand this it helps to imagine a physical realization of the objects.  So, let's imagine them as matchboxes with two strips of paper sticking out labelled A and B.  You can pull out one or the other and read the value, and it will say +1 or -1.  But as soon as you do, the matchbox catches fire and you destroy the remaining strip of paper.  Since the values read by the Bobs are correlated, clearly (or maybe not, as we shall see) there are just two possibilities:

1. Alice is writing the values on the strips of paper before sending out the matchboxes.
2. The matchboxes talk to eachother so the last one read knows what happened with the first two, and can quickly print the required value.

 I am including in #1 the situation where Alice doesn't literally write the values on the strips, but it is determined in advance what each Bob would get if he were to read A and what he would get if he were to read B.  So let's assume option #1 and work out the consequences.  Well, since Alice knows what each Bob would get if he were to read A we can label these $A_1, A_2, A_3$, and likewise the values each Bob would get if he were to read B can be labelled $B_1, B_2, B_3$.  Since we know the product of one A and 2 B's is always +1, and $B_i^2$ has to be +1, we can work out what the product of 3 As should be
$$
\begin{align}
A_1 A_2 A_3 &=A_1 A_2 A_3 B_1^2 B_2^2 B_3^2 \\
&=(A_1 B_2 B_3) (B_1 A_2 B_3) (B_1 B_2 A_3) \\
&=(+1)^3 \\
&=1
\end{align}
$$
But in fact $A_1 A_2 A_3$ is always -1.  This means we can eliminate option #1 and determine that #2 is true, i.e. that the matchboxes talk to eachother on being read and there is a causal link between the Bobs.  But here comes the shock: this experiment still produces the same result when the Bobs are a space-like interval apart.  That is to say, Bob1 is far enough away from Bob2 such that any signal carrying information about the value read by Bob1 would arrive after Bob2 has read his value, and likewise for any other pair of Bobs.  So we either have to ditch Einstein's postulate that no signal travels faster than light (and with it, all of relativity) or we have to eliminate option #2 as well.

It sounds like such an experiment cannot really be designed, but in fact it can!  (At least in principle, I am not sure whether this experiment has actually been performed, but these results are predicted by the mathematics of Quantum Mechanics which has so far passed every test thrown at it.)  For the benefit of the QM-savvy, the objects can be anything with two orthogonal states, for example electrons which can be spin up or spin down.  Let's stick with electrons, and let's call $\lvert\uparrow\rangle$ the spin up state, and $\lvert\downarrow\rangle$ the spin down, and $\lvert\uparrow\uparrow\uparrow\rangle$ the product state where all 3 are up.  Alice places the system in the entangled (non product) state $\frac{1}{\sqrt{2}}(\lvert\uparrow\uparrow\uparrow\rangle - \lvert\downarrow\downarrow\downarrow\rangle)$ and sends each Bob a single electron.  The A that each Bob can measure is $\sigma_x$, the spin along the x-axis, and the B is $\sigma_y$, the spin along the y-axis.

So, let's repeat the two mutually exclusive and - we thought - exhaustive options that we have eliminated, and let's abstract them a little bit while we're at it:

1. There are hidden variables that pre-determine the results of reading A or B
2. There are no hidden variables, but there is faster-than-light communication

What's the 3rd option...?

Viewed from the Many Worlds Interpretation, there is indeed a third possibility, one that doesn't require hidden variables, or faster than light communication.  Or, in fact, special non-physical powers possessed only by sentient beings.  It is simply that the system starts off in a superposition of product states (all three up and all three down) and ends up in a superposition of product states, such as "all Bobs read A and got -1" ($A_1= A_2= A_3=-1$), or "Bobs 1 & 2 read B and got -1 and Bob3 read A and got +1" ($B_1 = B_2 = -1, A_3 = +1$).  All that is needed to resolve the paradox is to accept that Bobs can be in superpositions of states, and not just electrons!