0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

In a nutshell, it's due to (1) interactions, and (2) the temporal duration of the experiment.

Basically, if the two qubits are separated by a distance $d$ and the experimental protocol takes place over a time interval of duration $T$ that's larger than several times $d/c$, then relativistic causality concerns do not rule out the possibility of interactions between the two qubits effecting a global quantum unitary.

This is what we mean by "far" when we talk about spacelike separations in an LOCC context: terms like "near" and "far" never make sense in isolation, and here the relevant yardstick is set by (the speed of light times) the time it takes to implement the operations. If you're very fast with your operations, then you'll be able to implement spacelike separations at smaller and smaller distances.

$\endgroup$
  • $\begingroup$ Im bit confused, maybe this question will clarify it for me. Suppose in a quantum computer I have 2 qubits which are very near and I want to apply a CNOT gate on both qubits (for ex. you do this by tuning some magnetic and electric interactions in a hamiltonian such that the correct unitary is applied). If I'm too slow then applying this gate will be impossible for any state on the qubits. As I get faster in the protocol I may get nearer and nearer to be able to apply the CNOT gate, yet I don understand how would the CNOT gate woul come out as I get faster, a explicit calculation would help me $\endgroup$ – Pam Aug 18 '18 at 9:47
  • $\begingroup$ You've got it backwards. If you want to go too fast, then the gate becomes impossible. There's nothing stopping you from implementing the gate as slowly as you want to, but the light travel time between the two locations does act as the minimal time it takes to perform any non-LOCC operation. $\endgroup$ – Emilio Pisanty Aug 24 '18 at 18:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.