I was reading about LOCC recently, wherein two parties (Alice and Bob) are only allowed to apply local unitaries in their corresponding qubits and communicate classicaly.
To motivate this situation, Alice and Bob are supposed to be in far away labs, so that its impossible to apply a global operation on both qubits since nature is local and there is a finite velocity (speed of light). Yet this poses a problem for me. If this is true then even if Alice's and Bob's qubits are VERY close (say a few micrometers or even less), then in principle they should still be able to only apply local operators on each qubit. One must then conclude that this local operations are the only possible operation on two qubit systems (or more systems that are bipartite). I imagine there is something very wrong with this reasoning but I can't understand why.
Any help with this problem will be greatly appreciated