# 1591: Bell's Theorem

## Explanation[edit]

Ponytail begins reading Bell's theorem to Cueball, who is standing 5 meters away. Bell's theorem, invented by the physicist John Stewart Bell, suggests that local hidden variables - that is, unknown properties of a system that are communicated via physical effects within the system's nearby surroundings - are not sufficient to fully explain quantum mechanics. This means that any complete description of quantum mechanics must necessarily include some *non-local* effect - some kind of influence that can be transmitted from some remote location not within the system's reach. Furthermore, that influence must necessarily travel *instantaneously* and does not obey the limit of the speed of light.

Cueball responds by misunderstanding this to mean that faster-than-light communication is actually possible. However, his misunderstanding occurs in 1 nanosecond. Since the speed of light in a vacuum is 299,792,458 meters per second, the light from Ponytail would have traveled only 30 centimeters, which means that Cueball has managed to misunderstand Bell's theorem faster than the speed of light - a feat that violates locality, just as the theorem predicts.

The punchline is that this is a special case known as Bell's Second Theorem: the idea that misunderstandings about what Bell's theorem means happen so readily that they actually violate the principle of locality.

This comic was published on October 16, 2015, five days before an article about the first-ever Loophole-free Bell's Theorem test was published in Nature magazine (DOI:10.1038/nature15759) (see also Bell test experiments). However, the paper was submitted almost two months earlier on the 24th of August and could most likely be found online before this comic was released. It was accepted by Nature already on the 28th of September, but was first published online October 21, 2015. Randall may very well have been aware of the imminent release of this paper, although it is peculiar that he did not wait until the paper was released. (This could potentially be a meta-joke, with the joke about Bell's Theorem being released before the paper about the relevant experiment was published)

Another way to state Bell's theorem is "No physical theory of (finitely many) local hidden variables can ever reproduce all of the predictions of quantum mechanics." It says that a theoretical treatment that divides the universe up into separate ("local") systems like this will always discard something about those systems' intercorrelations.

It is possible that there could be "global hidden variables" which share information across systems, perhaps by some manner of superluminal communication - however, this has unsettling philosophical implications such as superdeterminism, where the universe is essentially just reading off a script and no free will is possible. Needless to say, many people find this an unsatisfying resolution.

The preferred resolution of the paradox is not to insist (as early physicists did) that the universe's state is a collection of bits (classical information), but treat it as a collection of qubits (quantum information).

In quantum mechanics (QM), "measurement" is the process of allowing a small system to interact with its environment in a controlled way. The interaction allows information about the system's state to escape to the environment, producing an "observation." If the measurement apparatus is governed by classical mechanics (impossible in reality, but a very common simplification for the purposes of calculation), then the observation can be thought of as classical information, a bit (yes/no answer) in the simplest case. While the system may have been in any one of infinitely many states before the measurement (each a superposition of classical states), the fact that the measurement must leave it consistent with the classical result means that it can end up in only finitely many states afterwards. This is the "wave-function collapse" of early QM, popularized by Schrödinger's cat, but unrelated to the Heisenberg Uncertainty Principle, with which lay audiences often confuse it.

Modern quantum mechanics acknowledges that the environment is not classical, and that wave-function collapse happens by a (comparatively) gradual process called "decoherence," where information leaving the system is made up for by information coming from the environment that drives the system closer and closer to one of the finitely many states predicted by the simplified model above. If a "Schrödinger's cat" is in a half-and-half superposition of the states "dead" and "alive", when its liveness is measured, the ratios of "dead" and "alive" will shift rapidly towards (though not quite reach) 0 and 100% or 100 and 0%. For all but the shortest time scales, the cat's post-measurement state might as well be classical.

Entanglement is a situation where the future outcomes of two or more measurements that would be independent in a classical world are nonetheless correlated. For example, two widely separated electrons from one source could be in a state where, considered individually, each is in a superimposed spin-up/spin-down state, but if one is measured as spin-up, the other will necessarily be measured as spin-down. This is untroubling if the two electrons are modeled as a single system, but strange-seeming if we think of them as separate: how did the measurement of the first electron allow information from the environment around it affect the far-away second electron? It seems like the electrons are communicating, potentially at superluminal speeds, which would violate either relativity or causality. In actuality, there's a fairly simple proof (see below) that correlations from entanglement can't be used to communicate, and causality and relativity are safe. But that doesn't make the seemingly faster-than-light effects much less of a surprise.

One can try to address these concerns by considering 'local hidden variables', classical properties of a local system (like a single electron) that could have been observed but were not. For example, perhaps a classical part of the electrons' state lets them "agree" on a future classical state at the moment they are entangled, and then they just reveal that state in the future. But this becomes unwieldy: there are infinitely many possible future observations the electrons would have to agree on, and it seems difficult to do this without infinitely many local hidden variables.

The title text jokes about the No-Communication Theorem. The real theorem states that although determination of the state of one half of an entangled pair immediately determines that of the other half, however far away it may be, there's no way for the observer of the other half to see if he's the first to find out the state or whether it'd already been determined by the first observer. Thus, no information travels from one observer to the other.

Randall's version of the *No-Communication Theorem* states that no matter how you try to send information about this theorem (no communication about the No-Communication Theorem) then it cannot clear up the misunderstanding about Bell's Theorem quickly enough that any correct information (about Bell's theorem) has actually been transferred faster than light. So the conclusion is the same as the real No-Communication Theorem - faster-than-light signaling is not possible...

## Transcript[edit]

- [Ponytail, facing right, is holding a piece of paper with both hands. In a small frame breaking the top of the large frame is a caption:]
- t= 0 nanoseconds
- Ponytail: This is called Bell's Theorem. It was first–

- [A double-headed arrow points to Ponytail and then exits the frame crossing into the next frame where it points to Cueball. The arrow is split in two parts and in the center across the border of the two frames is a label:]
- 5 meters

- [Cueball, facing left. In a small frame breaking the top of the large frame is a caption:]
- t= 1 nanosecond
- Cueball: Wow, faster-than-light communication is possible!

- [Caption below the frames:]
- Bell's Second Theorem:
- Misunderstandings of Bell's Theorem
- happen so fast that they violate locality.

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# Discussion

I'm sure some people here have this memorised, but light travels just under 30 centimetres in a nanosecond. For our Metric-ally challenged friends, that's about one foot – so 5 metres takes around 16.67 nanoseconds. I leave the comic explanation to smarter people than me. Paddles (talk) 13:02, 16 October 2015 (UTC)

- I have seen Admiral Grace Hopper demonstrate this with approximately foot-long lengths of wire representing "light-nanoseconds". It's accurate to one part in 50 (although not as accurate as the one-part-in-1000 "30 centimeters" measurement). PsyMar (talk) 20:33, 16 October 2015 (UTC)
- The problem with that nifty rule-of-thumb is that it is technically correct, but practically useless. The 30cm/ns is for light
*in a vacuum*. For an electrical signal in a wire (or light in a fibre, for that matter) the effective speed is roughly 20cm/ns. -- Popup (talk)*(please sign your comments with ~~~~)*

- The problem with that nifty rule-of-thumb is that it is technically correct, but practically useless. The 30cm/ns is for light

The comic only shows that the two characters are 5m apart at chest level. What if there was a miniature wormhole or distortion in time in a separate area, making this seemingly "FTL" communication scientifically possible? Forrest (talk)14:19, 16 October 2015 (UTC)

For an explanation of Bell's theorem in the words of the man himself, and targeted at an educated lay audience, this is essential reading: https://cds.cern.ch/record/142461/files/198009299.pdf 162.158.35.36 16:22, 16 October 2015 (UTC) : Tim B posting as Anon

Wow, the explanation needs some explaining. Can the first part about quantum mechanics be simplified, moved, or have something clearer put in front of it? I don't feel up to the task, but the section is not very helpful. -DanB (talk) 17:32, 16 October 2015 (UTC)

- Yeah, the explanation isn't actually an explanation at all. Can someone who understands Bell's Theorem write an explanation for the joke in the comic? The current explanation appears to be a non sequitorial digression. I'm really curious as to what the actual joke is about. 108.162.249.155 04:20, 9 March 2016 (UTC)

In the widely separated electrons section, isn't it necessary that the two electrons measured be from the same source? If so, the explanation could use that small edit, but I'm not sure I'm remembering right. Miamiclay (talk) 05:35, 17 October 2015 (UTC)

- Yes.

I think this whole explanation is suffering from "Bell's second theorem".

Can anyone cite an experiment or proof that *altering* the state of one half of an entangled electron pair *after* they have been separated to a significant distance has any effect upon the other half? So far as I have learned, the two electrons in question are driven to opposite states by close proximity: When separated, they maintain cyclical synchrony until the state of one electron is measured. Environmentally induced state changes have not been shown to propagate between entangled particles after they are separated; They simply retain oppositional synchrony until disentangled by observation (or other interference). Any information derived was imparted at the point of entanglement, or during transit, or by measurement. Introducing new information (state change) to one half of an entangled pair after separation interrupts the synchronous effect, disrupting the entanglement. This is not useful from a communications standpoint.

- Correct, there is nothing that changes about the second particle when the first particle is measured

Nothing in quantum mechanics actually violates classical mechanics; Rather, quantum mechanics acknowledges that our ability to measure a near-infinite (but still finite) set of variables is limited by the effect of our own observation & by our inability to quantify all relevant variables prior to comparison. Thus "quantum uncertainty" & wave function collapse are merely an admission that any data set is necessarily incomplete, while reserving the possibility of predicting deterministic outcomes by reasoned observation of the limited data available.

At least, that's what the cat told me. 108.162.221.47 06:54, 17 October 2015 (UTC)

- That is exactly what Bell's theorem states and what the experiments behind it showed. It is a bit technical, but the best layman description I have seen was on Ars Technica: http://arstechnica.com/science/2010/01/a-tale-of-two-qubits-how-quantum-computers-work/

162.158.92.91 09:41, 17 October 2015 (UTC)

The first rule of the No Communication Theory is that you don't talk about the No Communication Theory. -Pennpenn 108.162.250.162 22:44, 18 October 2015 (UTC)

*"This means that any complete description..."* - isn't this exactly the misunderstanding the comic is making fun of? I don't think one can make such an inference without first unscientifically presupposing some interpretations of QM to be correct.