I wanted to understand Rob Spekkens’ self-described lonely view that the contextual aspect of quantum mechanics is more important than the non-local aspect. Although I like to think I know a thing or two about the foundations of quantum mechanics, I’m embarrassingly unfamiliar with the discussion surrounding contextuality. 90% of my understanding is comes from this famous explanation by David Bacon at his old blog. (Non-experts should definitely take the time to read that nice little post.) What follows are my thoughts before diving into the literature.

I find the map-territory distinction very important for thinking about this. Bell’s theorem isn’t a theorem about quantum mechanics (QM) per se, it’s a theorem about locally realistic theories. It says if the universe satisfies certain very reasonable assumption, then it will behave in a certain manner. We observe that it doesn’t behave in this manner, therefore the universe doesn’t satisfy those assumption. The only reason that QM come into it is that QM correctly predicts the misbehavior, whereas classical mechanics does not (since classical mechanics satisfies the assumptions).

Now, if you’re comfortable writing down a unitarily evolving density matrix of macroscopic systems, then the *mechanism* by which QM is able to misbehave is actually fairly transparent. Write down an initial state, evolve it, and behold: the wavefunction is a sum of branches of macroscopically distinct outcomes with the appropriate statistics (assuming the Born rule). The importance of Bell’s Theorem is *not* that it shows that QM is weird, it’s that it shows that the *universe* is weird. After all, we *knew* that the QM formalism violated all sorts of our intuitions: entanglement, Heisenberg uncertainty, wave-particle duality, etc.; we didn’t need Bell’s theorem to tell us QM was strange. But the outstanding question was whether the weirdness in the formalism of QM actually reflected a weirdness in the universe, or whether that was just an artifact of our incomplete understanding. And indeed, the EPR paper was trying to pin down a weirdness in QM which the authors believed to *not* reflect reality but instead would be exorcised by a successor theory. In other words, Bell’s theorem showed that the territory—not just the map—was weird.

As I understand it, contextuality is the property of quantum mechanics that says that the results of measurements are not pre-existing properties that are discovered, but rather do not become definite until probed. The evidence for this is that, for certain experiments in QM, the outcomes of different sets of measurements (as in, different pairs of triplets of measureable properties) will not have a consistent explanation in terms of individual properties.

This is certainly a good reminder that classical intuition can lead you astray, and it’s certainly a good illustration of what exactly measurements are in the QM formalism. *But it doesn’t tell us anything about the universe*. Here’s where I think the key difference between it and Bell’s theorem lies: we have a clear, unambiguous idea of spacetime points. Yes, there might be some confusion at the Planck level, but everyone and their mother agrees when two macroscopically separated points are space-like or time-like. We know that the detector over here is not on occupying the same physical space as the detector over there.

On the other hand, no such clarity exists for a “measurement of a property”. Measurements rely on a chain of causality running from the measured system to our minds, and it is not at all clear when two measurements at different times are measuring the same property unless we agree on what happens along the chain. (And in quantum mechanics, the chain fades into the microscopic.) We can all agree about the physical signal produced at the end of measurement (e.g. the green light lit up), but no such agreement exists about what property was measured. Without this agreement, there are no experiments to be run to prove that the universe, rather than just QM, is contextual^{ a }^{ b }.

In other words, the agreement on spacetime points is what makes Bell’s theorem a test of the *universe* rather than just our QM model of it. Now, of course, the philosopher of science may point out that all observations are “theory laden“, so nothing is a pure test of just the universe, unbound from basic assumptions (e.g. that radical skepticism is false, and that we are not a brain in a vat). But the key part of Bell’s theorem is that no one actually disputes spacetime, whereas the inferential measurement chain cannot be separated from QM. Bell’s theorem is built on much firmer foundations, where firmness is measured by my subjective feeling of confidence and the objective number of people who agree.

Is there something more to it than that? Well, it may be important that I’m an “Everettian with reservations”^{ c }. I never think of quantum systems as having observable-represented properties (like position) independent of their wavefunction. I just think “OK, there is some wavefunction describing a system interacting with an apparatus. What are the possible macroscopic outcomes, and what are the respective probabilities?” Measurement-like interactions are just one of many interactions possible for quantum systems described by wavefunctions.

On the other hand, if you had a view of the universe where measurements played a fundamental role, you might be surprised by contextuality. Because if your measurements are fundamental operations, then it is intensely important to understand what exactly they do. And, in this case, you might be surprised to find that your fundamental measurements are really more akin to “ask the universe a question, to which the universe may answer in various ways based on context” than to “inspect this pre-existing degree of freedom”.^{ d }

I’m not sure yet whether my nonchalance with contextuality is compatible with the fact that I find Spekken’s toy theory deeply insightful. I think Spekken’s models is telling us something very important about QM and, in fact, shows that almost all the weird parts (including contextuality) could appear in an essentially classical universe. The crucial exception is, of course, Bell violations!

Now, it seems plausible to me that (1) Bell’s theorem is the single best identification of what’s weird about our universe but (2) contextuality and Spekken’s toy model will actually get us closer to discovering how to crack open QM, i.e. will lead us to whatever successor^{ e } theory leaves us feeling less confused. I’m not convinced of (2), but I think it’s compatible with (1).

### Footnotes

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- As described by places like the Stanford Encyclopedia of Philosophy, there was a significant debate about whether the Kochen-Specker was empirically testable, and the general conclusion is that it was not. I have not read enough of the relevant literature to know if the explanation I give in this blog post is compatible with that previous discussion. I could definitely be wrong in my thinking.↵
- I still have to read Rob’s paper “Contextuality for preparations, transformations, and unsharp measurements” to see if he says something that would change my stance.↵
- I think it’s silly to ascribe just as much “reality” to the wavefunction as to a rock. Our knowledge of the wavefunction is based on a far longer, more uncertain, and more indirect chain of inferences than our knowledge of the rock. And the wavefunction resists confirming measurements in a baked-in way that a rock does not. Nonetheless, a unitarily evolving wavefunction I can understand without ambiguity (even in the context of quantum cosmology) whereas the same cannot be said for operational approaches that assume classical objects apriori or use measurements as fundamental operations.↵
- A more charitable account of quantum operationalism might not consider measurements as fundamental mathematically rigorous operations, but instead are less specifically specified set of experiment preparation procedures. This to me would obviously be unacceptably vague, since it would not allows us to ask and answer questions about, say, quantum properties of the CMB.↵
- Here, “successor” just means we use it to understand the universe in lieu of QM. It may involve unambiguous departures from QM in experimental predictions, or it might be different in a more subtle way by changing what we’re even talking about.↵

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