There is a much simpler argument to be made here: send a stream of entangled particles through a polarizing beam splitter. At each branch, put another polarizing beam splitter oriented at 45 degrees to the previous one. You can continue this indefinitely to split the beam an arbitrary number of times. QM predicts that entangled particles must remain correlated throughout the splitting no matter how many splits there are. If there were only a finite number of hidden variables the correlation would have to stop after a finite number of splits.
You're missing the point. Locality, topology, and even geometry are irrelevant. What matters is how many bits of information you would need to specify the path that the photon takes. And the answer is: you need an infinite number of bits. (OK, well, actually you need an unbounded number of bits, which is not quite the same thing. But it's equally implausible.)
So if you use an infinite amount of matter to create an infinite number of beam splitters, an infinite amount of energy to move them to an infinite number of positions (which I guess you keep track of using a computer with infinite memory), over (I guess again) an infinite amount of time, then...
... it turns out that you also need an infinite number of hidden variables for the resulting setup to work according to QM?
I mean, your argument supposes that the system can't have that many bits -- but doesn't your device to generate arbitrary bits require more bits to be generated than that amount?
Your argument sounds like "5 bits is too many bits to have in a system, and my 10 bit device can generate 5 bits!"
I guess? But that doesn't really seem interesting to me: impossible outcomes from impossible to construct situations don't matter.
Also, any value which can take on an arbitrary real (or even rational) number has arbitrary storage capcity -- and so far as I know, lots of quantum values do.
"Now an international collaboration led by Adán Cabello has invoked a fundamental thermodynamics result, the Landauer erasure principle, to show that systems in hidden-variable theories must have an infinite memory to be compatible with quantum mechanics."
All I'm saying is that you don't need thermodynamics to show this. Conservation laws are enough.
My point was your argument doesn't hold, because you cant actually construct a situation in which the system doesnt already have sufficient storage capacity: the inability to require those bits without already having enough bits in the system to store them is pretty important. I don't believe that it can require QM to account for arbitrary memory, without presupposing we have it to construct the device. Hence, impossible constructions dont matter.
I have some similar objections to the summary of their paper, but I haven't had time to read it yet. It's possible they address some of them.
In particular, measuring from an arbitrary location and an arbitrary number of times sounds like the introduction of the infinity by assumption, rather than by consequence.
I think you're missing the point. It is not that all that information exists, it is that all that information has to exist in a single particle. This is not impossible, but it is implausible.
Correction: it has to be accessible to a single particle, but the information could be non-localized across the whole system, since your particle is itself fairly non-localized, and arguably has weakly interacted with the various devices along the way, meaning that Im not sure we can fully disentangle the wave-equation for the photon from the equations for the splitters. This could easily be the photon "borrowing" bits from the splitter, by sharing a non-local value with it, which carries the split path information for both.
Which is just to say that the system (in full) has to be able to contain a number of bits sufficient for that, which your setup does.
> Im not sure we can fully disentangle the wave-equation for the photon from the equations for the splitters
That doesn't matter. The two sides of the experiment are space-like separated, and the splitters on each side are light-like separated. There is no place the information can possibly be other than in the photons.
...except non-local variables. Which is why I keep mentioning non-locality.
I don't think words are working and I'm not sure I understand your point fully. Could you draw a diagram? Specifically, the apparatus and where you think the requirement for information occurs.
I really do want to understand your point, because it's an interesting argument, even if I'm not sure I agree.
Could you point out what you think that has to do with locality?
It mentions it once, to assert it as a desired axiom (and incorrectly assert that QM can be local, which it can't be). I also would argue that GR supports non-local theories well enough, in that the actual geometry can appear macro-Euclidean (or other nice space) while containing micro-bridges which break the locality structure for the "nice" macro space.
A non-local value is any value which can impact things at super-luminal speed (generally, instantaneously). Of course, I would argue that the non-locality is only apparent, and we're simply seeing the shape of things very wrong because of how brains work, the information reaches us, etc.
> A non-local value is any value which can impact things at super-luminal speed
Ah. Those are ruled out by relativity. If you can send information faster than light, then you can send information backwards in time and violate causality.
> Cabello and colleagues show, however, that if an experimenter is free to make spin measurements anywhere in the xz-plane, the heat generated per measurement is unbounded—obviously, an unphysical result.
First thing that popped into my head was "ultraviolet catastrophe" -- and saw that one of the scitation.org commentors had beaten me to punch.
Note the first assumption: "The choice of which measurement is performed can be made randomly and independently of the system under observation". In the Physics Today article, this becomes "if an experimenter is free to make spin measurements anywhere".
Either way, the conclusion that "systems in hidden-variable theories must have an infinite memory" only holds if you assume free will.
