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Imagine Alice and Bob have two pairs of entangled particles, pair A, and pair B. Imagine they have agreed beforehand that pair A measurement preceding pair B measurement constitutes a bit with value 1 and pair B measurement preceding pair A measurement represents a 0 value bit. Would they not be able to use this predetermined order to transmit at least one bit information instantaneously across large distances?

Where is the flaw in this chain of reasoning?

Tobias Fünke
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Abhishek Nair
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The quantum mechanical expectation value (prediction) is generally independent of order when there is entanglement. Whether Alice measured first or not has no observable difference to the outcomes as far as anyone knows. The outcomes Alice alone sees are always random. Same with Bob. Not much information to be gained from looking at a random stream of bits.

DrChinese
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Suppose Alice measures first. She measures A then B to send a 1.

What does Bob do? He measures both in either order. He gets answers. How does he know the order Alice made her measurements? Or if she has meade measurements?

mmesser314
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If A before B, then one of them, say Alice, gets 1 bit. Bob doesn't know any thing. He could measure in any order. Only if Alice sends a message to Bob telling him that she got 1 bit, Bob will then know what to do and will be able to predict what he's going to get. But notice, this was only due to a conventional communication that took place no faster than the speed of light.

Anky Physics
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Alice and Bob are living in a relativistic universe. Who measures first is observer dependent. This is usually called the Andromeda paradox. Since all observers have to agree with each other about the outcome of those measurements, the order of events can not convey any kind of information, even if it can be correlated. The main problem with the discussion of quantum mechanics at the level of entanglement is that it drags the (incorrect) structure of Galilean spacetime into the picture by analyzing it exclusively in the center of mass system of the two measurements.

I am not a theorist, but intuitively it seems to me that quantum mechanics is a relativistic phenomenon. There can be no self-consistent non-relativistic version of it and if we look at the endless and fruitless discussions of these phenomena at the level of non-relativistic theory nature seems to back that notion up.

FlatterMann
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  • Alice and Bob could be at rest one relative to the other… – ZeroTheHero Jan 16 '24 at 02:56
  • @ZeroTheHero I am not talking about the relative motion of A and B. I am talking about ANY external observer. The measurement results of A and B are invariant (since they are just counts they are scalars). It's the statement that A measures before B that is meaningless in relativity. It is only meaningful in a Galilean universe, which does not exist. In QFT all of this is accounted for by the fact that wave functions are fully symmetric or antisymmetric. Energy at A is treated exactly the same as energy at B. – FlatterMann Jan 16 '24 at 07:05
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    The issue of relativity plays no role here. Even in the non-relativistic formulation of QM, the no-communication theorem shows that entanglement cannot be exploited for FTL communication. In QFT the local (or micro-)causality condition of fields also establishes that. But you are right, there it also important that if both measurements are space-like separated, the predictions must be independent of the order (which they are, again due to the same condition on the fields/observables). – Tobias Fünke Jan 16 '24 at 08:52
  • I wasn't trying to say that non-relativistic quantum mechanics makes an FTL prediction. It's the incorrect analysis in the center of mass system that makes false assumptions about the meaning of a two-quantum state (as a spatial state rather than a Hilbert-space element) that does. The point is that a relativistic analysis makes it very obvious why such a thing simply can not happen. It explains more about the structure of quantum mechanics in addition, but that's for another question. – FlatterMann Jan 16 '24 at 10:18
  • I don't understand and I don't see any assumptions the OP is doing on the center of mass system or so. The structure of any quantum theory a prori does not know anything about relativistics. Even in non-relativistic QFT you need, as I said, the postulate of micro-causality. In fact, this axiom is motivated by the fact that you want the no-communication theorem (+ relativistic invariance) to hold. A nice discussion is found in C. Beck's book on local quantum measurements. – Tobias Fünke Jan 16 '24 at 10:33
  • The OP assumes an absolute event order. Nature does not. The structure of quantum theory doesn't just drop out of nowhere. It seems that nature does not have a choice in the matter, either, if relativity holds. How you want to axiomatize this is up to you, of course, but I prefer to start with as few axioms as possible and I believe that relativity is the preferred axiom. After that the structure of QFT seems to be fairly strongly limited as representations of the Poincare group. The structure of non-relativistic quantum mechanics follows from that as a limiting case. – FlatterMann Jan 16 '24 at 10:37
  • Well, in non-relativistic QM (or classical mechanics) there is an absolute order. Of course, this violates what we know from relativity, but this was not my point to begin with (although I agree with your point, if I understood it correctly, that if one starts with a non-relativistic theory, one should not expect it to be consistent with it). Despite that, I don't think that a unitary representation of the Poincare group (roughly speaking) in relativistic QFT alone suffices to get no-communication/order invariance, though. You still need micro-causality. – Tobias Fünke Jan 16 '24 at 11:09
  • @TobiasFünke Let's take this to chat. – FlatterMann Jan 16 '24 at 11:12
  • Let us continue this discussion in chat. – Tobias Fünke Jan 16 '24 at 11:13