In the last post, we took a non-standard look at uncertainty relations. This was the crucial picture:
We’ll call a setup like this a network box. It has exactly the same structure as a certain set of possible experiments on a quantum spin-one system. Given an input (a state, represented here by a hex number) and a choice of triangle (experiment, picked out by the pointer position), exactly one of the lights at the vertices of the triangle comes on when you push the button (one result out of three possible.) We focused on two constraints:
(1) If three vertices form a triangle, their associated probabilities (i.e., the probabilities for the lights to come in a relevant experiment) sum to 1.
(2) If a vertex belongs to more than one triangle, its probability doesn’t depend on which triangle we associate it with.
As we saw, (1) and (2) give us an uncertainty relation:
(3) p(a) + p(m) <= 3/2.
Here we add: (3) tells us that p(a) and p(m) can’t both be 1:
(4) p(a) = 1 if and only if p(m) ≠ 1.
We’ll need (4) later. This post takes a step beyond Pitowsky. The picture we need now is a bit different:
Imagine a source that produces network boxes in pairs. We’ve left out the input screen because the joint state is built in at the source. Once the pairs leave the source, they aren’t connected in any ordinary physical way: no wires, no radio signals… But just as before, if you pick a triangle and push the button, one of the triangle’s vertices light up.
(1) and (2) still hold and so do (3) and (4) too. But this time, the probabilities for the individual network boxes are simple: each outcome has a probability of 1/3. For example: if you set the dial to 7 and push the button, each of the vertices j, k, and l has a 1/3 probability of lighting up.
Suppose Alice has the left-side box and Bob the right-side. They pick settings that overlap. For example: Alice picks 5 and Bob picks 7. The triangles overlap on vertex: k. The new and important bit is this: quantum theory says that either k will light up for both Alice and Bob or for neither. This extends to the case where they pick the same setting. For example: if Alice and Bob both pick 1, the results will either be (a,a) or (b,b) or (c,c).
There’s a special state of two quantum spin-one systems that predicts correlations exactly like this. We do need to add a qualification: the correlations break if Alice or Bob chooses a new setting that doesn’t include the old vertex and runs the experiment again on the same system. We’ll make a stronger assumption to keep the presentation simple: each network box only works once. To keep experimenting, Alice and Bob need to go back to the source and get a new pair of network boxes. We’ll assume there’s no shortage, and in fact a realistic quantum experiment would almost certainly work this way.
Suppose you want to explain how the results for pairs manage to line up as we’ve described. One hypothesis is that the boxes are connected by a causal link of some sort, but we’re assuming that no obvious such link can be found. Taking that seriously, EPR-style reasoning has a lot of appeal. If Alice picks 1 and gets a, she can predict apparently without influencing Bob’s box that he’ll also get a if he picks 1 or if he picks 4. EPR would say that there must be some “element of reality” that’s already built into Bob’s box and determines his result. The simple explanation seems to be: every pair of boxes comes programmed so that experimental results are already determined, with the same wiring for each box in a pair. For example: a pair might be wired so that the light in the red rectangle comes on if you pick a triangle that contains it:
The appearance of probabilities strictly between 0 and 1 (1/3 in this case) at the surface level is just a matter of sampling: box pairs are always wired alike, but there are many ways to wire a pair that would produce perfect correlations of the sort we’ve described.
This is an appealing story. If it succeeds, there won’t be any mystery about the correlations. They’ll just be a result of the way the boxes were wired at the source. The quantum analogue would be that the correlated quantum experimental results aren’t a matter of spooky action at a distance; they reflect what EPR would call “elements of reality” that the quantum systems already had when they left the source. But given what we’ve already said, it leads to a prediction.
Let’s use bold-face p for the “hidden” 0-1 probabilities that we hope will explain the correlations. If the diagrams represent the fundamental structure of the “wiring,” then (4) still holds for these probabilities. We can never have p(a) = 1 and p(m) = 1. If p(a) = 1, then p(m) has to be 0. But suppose Alice picks setting 4 and Bob picks setting 3. If our story is correct, there could never be a case where a comes on for Bob and m comes on for Alice. (4) rules that out. But if our networks are good models for quantum mechanics then this will happen sometimes. In fact, depending on the details the probability that it happens could be as high as 1/9. Our story can’t explain the correlations; not for our imaginary boxes and not for real-world quantum systems.
This is a different way of getting at what Bell’s theorem gets at. It’s also testable in principle, thought the experiment would be difficult in practice. The quantum states that produce behavior like this are among the ones we call “entangled,” and we have a simple demonstration that there’s no straightforward, “local” explanation for entanglement.
All this might suggest that the real-life quantum experiment would be a demonstration that what Alice does to her system somehow influences Bob’s system and vice-versa. In fact, I think it would be a mistake to jump to that conclusion, though it’s a mistake that many people (including many physicists) make. But that’s a whole ‘nother subject. For now, we can see this case as another confirmation of the power of looking at quantum probabilities combinatorially.