# 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?

• 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. – user1588914 Apr 2 '18 at 10:56

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.
• @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. – knzhou Apr 5 '18 at 15:37