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So non-relativistic quantum mechanics prevents superluminal communication. Given a bipartite state $\rho_{AB}$, the reduced density operators for systems $A$ and $B$ are given by the partial traces

$$ \rho_A = \mathrm{tr}_B(\rho), $$ $$ \rho_B = \mathrm{tr}_A(\rho). $$

Superluminal communication means that a local operation (measurement) on one part of the system would result in an instantaneous and measurable change of the other part of the system. It can be shown that a local operation on system $A$, given by acting $M \otimes I_B $ on $\rho$, would change $\rho_A$ but would not change $\rho_B$. Is it just a coincidence that superluminal communication is prevented? Nowhere have I incorporated relativity into my formalism, it is purely non-relativistic so there isn't a restriction on how fast things can travel. Surely I am still allowed to superluminally communicate by sending a particle from $A$ to $B$ by exceeding the speed of light?

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    $\begingroup$ I believe that what you're showing is that local operations alone on a bipartite state (entangled or not) are not sufficient for communication of any kind. Therefore, you're not proving that superluminal communication is not allowed, but that some sort of particle/energy exchange is necessary. I'm not aware of any purely quantum limits on particle exchange speeds, but it's an interesting question. $\endgroup$ – user1588914 Apr 2 '18 at 10:56
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No, nonrelativistic QM allows superluminal communication. For example, if I perfectly measure the position of a particle, it will instantly spread out over all space immediately afterward, faster than the speed of light. This can be used to signal to a distant observer. If the observer was along the $z$-axis I could even shoot the particle at them in a focused beam by measuring only the $z$-position. This issue is solved in relativistic quantum field theory by constructing fields which commute at spacelike separations.

What you've shown is that you can't transmit information through entanglement by local operations. But nothing about this derivation has anything to do with relativity; it also forbids transmitting information slower than the speed of light. It's just a constraint on a particular kind of communication.

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  • $\begingroup$ I don't understand how your answer is compatible with the non-signaling principle, which is obeyed by quantum mechanics. (e.g. journals.aps.org/pra/pdf/10.1103/PhysRevA.71.022101 , en.wikipedia.org/wiki/No-communication_theorem). $\endgroup$ – Wolpertinger Apr 5 '18 at 15:11
  • $\begingroup$ The principle says exactly the opposite of your second sentence, namely that measurement on a quantum state does not allow for superluminal communication. Of course you could (in principle) have equations of motion in non-relativistic QM which allow superluminal propagation of the particles (and thus the information in the quantum state), but that is usually assumed not to be the case on physical grounds. $\endgroup$ – Wolpertinger Apr 5 '18 at 15:13
  • $\begingroup$ @Wolpertinger There's no contradiction here. As I stated in the second paragraph, the no-signaling theorem applies to one specific kind of signaling, i.e. transmitting information by measurement via entanglement. Of course a nonrelativistic theory can violate relativity in plenty of other ways; there is no $c$ limit in the Schrodinger equation. $\endgroup$ – knzhou Apr 5 '18 at 15:37
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    $\begingroup$ don't get me wrong, I'm happy with most of your answer and also agree with the main point of your that you reiterated in the comment (that there is no c limit in the Schrodinger equation). What I disagree with is "For example, if I perfectly measure the position of a particle, it will instantly spread out over all space immediately afterward, faster than the speed of light. This can be used to signal to a distant observer." Because that is, at least in this formulation, exactly what no-signaling forbids and not one of the "plenty other ways", which i would be ok with ;) $\endgroup$ – Wolpertinger Apr 5 '18 at 15:44
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It is not just about superluminal communication. Actually the facts you state mean, that local operations do not allow communication at all, no matter what speed. For communication time evolution is needed.

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