Kochen-Specker Theorem explained

I previously explained Bell’s Theorem, which is a “no go” theorem of quantum mechanics. In brief, Bell’s Theorem proved in 1964 that any hidden variable interpretation of quantum mechanics must be nonlocal.

Of course, you may be thinking, maybe the world just is nonlocal, and that hidden information is being passed around faster than light. Unfortunately, there’s another major theorem which makes hidden variable theories even more unpalatable. In 1966-1967, the Kochen-Specker Theorem proved that any hidden variable interpretation must be contextual.

To understand the meaning of “contextual”, suppose we have a quantum cat, and the cat has many possible states. It could be awake or asleep. It could be happy or unhappy. Or the cat could be none of those things because it is dead. Now suppose there are two possible measurements, which answer the following questions:

(1) Is the cat awake, asleep, or dead?
(2) Is the cat happy, unhappy, or dead?

This is a quantum cat, so you can only choose one of the two measurements. However, even if you can’t make both measurements experimentally, you might reasonably expect that the outcomes of the two measurements are related to each other. Specifically, if measurement (1) would find a dead cat, then so would measurement (2), and vice versa. This assumption is called non-contextuality. This cannot be true of hidden variable interpretations of quantum mechanics! Such theories must be contextual.

Figure 1: ambiguous cat Fig. 1: Cat of ambiguous state. Credit: Visentico / Sento

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Bell’s Theorem explained

In 2009, I wrote an explanation of Bell’s Theorem that could be understood by popular audiences. I wanted to repost it, but ended up rewriting it completely.

Although the predictions of quantum theory are well-understood, its interpretation is famously difficult. Because quantum mechanics only makes probabilistic predictions, many people have desired a “hidden variable” interpretation, where quantum objects have definite states, despite appearances to the contrary. But hidden variable interpretations are generally not accepted.

It is certainly possible to create a hidden variable interpretation that agrees with the all the predictions of quantum theory, and de Broglie-Bohm theory is an example of such a interpretation. However, de Broglie-Bohm has a number of unsatisfying properties. Indeed there are a few theorems that prove that any hidden variable interpretation must have unsatisfying properties.

The most important of these is Bell’s Theorem, formulated in 1964. What follows is an explanation of the thought experiment, the mathematical proof, and its implications.

The setup

Bell’s Theorem considers a particular thought experiment, in which two electrons are emitted simultaneously from a single source in opposite directions. This source emits electrons that are “entangled”, meaning that their quantum states are correlated with one another. If you perform the same measurement on both electrons, both measurements will always produce the same result.¹
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Origami: Square Star and other tesselations

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Square Star, by Ekaterina Lukasheva. The squares are on the side, not visible in this photo. Ekaterina has her own fancier photos here.

The past month has been astonishingly productive, in terms of origami. I discovered that there was a nearby origami convention so of course I had to go. Most people were doing one-piece origami, so of course I ended up trying a lot of one-piece origami myself.

I was, however, pleased to see some modular origami representation, and in particular there was Ekaterina Lukasheva, of Kusudame.me. She gave a presentation on the connection between modular origami and origami tessellations. And afterwards, as a demonstration of principle, she showed people how to make the Square Star, shown above.

I think perhaps few people understood her talk, but as someone who is interested in the design of both modular origami and origami tessellations, I for one found it inspiring. Further discussion and origami below the fold.

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What is meaning?

PZ Myers has an interesting post discussing the motivations of people who join ISIS, and which attributes similar motivations to the alt-right. In short, it’s about people “who feel a lack of significance in their lives”, people “who felt culturally homeless”. People start “from vague dissatisfaction, and desire for social status and sexual success” and become radicalized.

For the most part, I feel powerless to do anything about the radicalization of men. The world is big and incomprehensible–and also I live in California. So the reader may forgive me if I turn this more introspective.

What even is this desire for “significance”? I feel that we in the atheist community have been discussing it for years, usually in the context of discussing the appeal of religion, and calling for the atheist community to fill the void that religion has left. If PZ is correct, some of those atheists went on to fulfill that need by joining the alt-right. But for all that discussion of “meaning”, I don’t think I understand it. I don’t know if this is something I feel myself.
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New FTB bloggers

I said back in October I would plug all the new bloggers on FTB, and then I forgot about it. And now I have remembered.

I Have Forgiven Jesus
Intransitive
Small Town Deviant

Note that FTB accepts applications for new bloggers, and reviews them quarterly. If you are interested in applying, the next review is on January 17. Please see this page for details on how to apply.

I had postponed posting these links because I wanted to give the new bloggers a chance to settle in, but one of the new bloggers (not linked) apparently never settled in. Sigh…

Expert caution, amateur engagement

In an earlier post, I talked about whether it is appropriate to judge historical people using today’s standards. I was surprised that commenters were so opinionated on an issue of arguably little importance.

On the “anti” side, multiple people argued that to truly understand history, you shouldn’t be so judgmental about it. But the thing is, I am not a historian, so why should I act like one? I do not perform any original historical research. The only way I might ever teach history is by sharing stuff I learned from Wikipedia or news articles. If I were to withhold judgment on historical people, I would not learn more about history, I would learn less because I would be less engaged.

Let’s switch to talking about my area of expertise, physics. I am a professional physicist, and most of my readers merely have an amateur interest in it. We have different attitudes towards physics, as well we should.
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