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The Set-Up

Let's say we want to entangle two qubits $\phi_a,\phi_b$ (at locations $a$ and $b$ respectively) that are spatially separated by distance $d$ (in natural units) at a given instance of time. We are allowed to use additional qubits to get this task done but the additional qubits also cannot be entangled at the beginning of the procedure, i.e., if we are to use $n$ additional qubits in this process, the initial state of the system will be of the form $\vert{\phi_a}\rangle\otimes\vert\phi_b\rangle\otimes_{i=1}^{n}\vert{\psi_i}\rangle$ where $\vert\psi_i\rangle\in S^2$ is the state of the $i^{\rm th}$ additional qubit. We are allowed to use both classical and quantum channels. I'm interested in figuring out as to what is the minimum amount of time in which we can get the two qubits, $\phi_a$ and $\phi_b$, entangled?

A Candidate Optimal Protocol

I can think of a simple way of entangling them in $d/2$ natural units of time:

  • Step 1: Start with two additional unentangled qubits: $\psi_1,\psi_2$ at the midpoint between $a$ and $b$.
  • Step 2: Create a Bell pair out of the local pair of qubits: $(\psi_1, \psi_2)$.
  • Step 3: Send $\psi_1$ to location $a$ using a quantum channel. Similarly, and simultaneously, send $\psi_2$ to location $b$ using another quantum channel.
  • Step 4: Swap the states of $\psi_1$ and $\phi_1$. Similarly, swap the states of $\psi_2$ and $\phi_2$.

Thus, we have transferred the entanglement between the pair $(\psi_1,\psi_2)$ to that between the pair $(\phi_1,\phi_2)$. The only step that takes time here is sending qubits over the quantum channels and the time taken is $d/2$ natural units of time.

But Is It Optimal?

Can we do better in terms of time? I have a gut-feeling that we cannot but I can't exactly put my finger on why -- after all, since entanglement cannot be used for communication, causality won't be violated even if we were to be able to establish instantaneous entanglement over spacelike separated qubits. However, I feel that $d/2$ should be the limit here due to the fact that LOCC cannot create new entanglement. Although, I can't quite figure out is how to use this fact about LOCC to properly prove that it would take at least $d/2$ natural units of time to establish entanglement between qubits separated by $d$ natural units of distance if we start out with zero ebits in our protocol.

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You can parallelise your proposal. Imagine you have $k+1$ points along a line where points 0 and $k$ are locations $a$ and $b$. Let $k$ be even for the sake of argument. At each even-numbered point $i$, a Bell pair is generated and the two qubits are sent to sites $i\pm 1$. (For $i=0,k$, the second qubit is sent to $\phi_a,\phi_b$.) Then, at each odd site, perform a teleportation of one qubit through the other Bell pair. The net effect is that you've got a Bell pair between $\phi_a$ and $\phi_b$. The time required is $2d/k$, which you can make arbitrarily small. I'm assuming that the time to send classical messages for the teleportation is negligible compared to the qubit transport time.

Just to explain why this is OK: we're messing a bit with the notion of LOCC by introducing all these way stations where something can be done. If you really only have locations $a$ and $b$, then it'll take you time $d$ because one has to send one half of an ebit to the other.

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  • $\begingroup$ Hi, thanks for the answer. I'm not sure I understand your protocol correctly. I've tried to illustrate the protocol schematically for $k=3$: Imgur Link. If this is indeed the protocol, I'm not sure why it's a valid move to ignore the time required to send classical signals -- it's of the same order as the time required to send qubits. One more confusion: if I ignore the time taken for classical communication, I think I'd get $d/4$ as the time needed to complete the procedure in the case of $k=3$. Your formula suggests it should be $2d$ ($>d$?!). Thanks a lot! :) $\endgroup$ Jul 18, 2022 at 12:28
  • $\begingroup$ Yes, that picture's right, just remember that you can do all the teleportations in parallel! $\endgroup$
    – DaftWullie
    Jul 18, 2022 at 13:16
  • $\begingroup$ The classical signals have to travel twice that distance that the qubits do, so the scaling is essentially the same. But classical messages can travel at the speed of light, which I'm assuming is a much higher speed than qubit transport, so the contribution of classical messaging time is trivial (but you can add it in if you want). $\endgroup$
    – DaftWullie
    Jul 18, 2022 at 13:18
  • $\begingroup$ Yes, I was assuming all communications to be at the speed of light. For example, in my protocol, there is communication over only quantum channel(s) and I take the time to be $d/2$ in natural units. So, assuming all communications to be at the speed of light, $d/2$ would be the limit? $\endgroup$ Jul 18, 2022 at 13:21
  • $\begingroup$ Ah yes, if you're forcing me to account for the travel time of the classical communication at the same cost as the quantum communication, then yes, that'll be the limit. I believe that while all my teleportations can be done in parallel, the classical messages still have to get through to one or other end of the chain. $\endgroup$
    – DaftWullie
    Jul 18, 2022 at 13:28

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