Is it a coincidence that non-relativistic quantum mechanics prevents superluminal communication? 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?
 A: 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.
A: 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.
