# Is there a non-deterministic protocol for entanglement generation between distant parties?

I'm aware that one can imperfectly clone entanglement that's shared between two parties (i.e. Bell pairs) using deterministic quantum cloning machines to produce two, lower fidelity entangled states.

What I want to know is, does there exist some strategy to non-deterministically generate entanglement between two distant parties? In other words if Alice and Bob have a Bell pair between them, is there some LOCC strategy they can do that will either create another Bell pair of the same fidelity, or fail wiith some probability.

No, any such protocol would violate the holevo bound (1 bit of communication per 1 sent qubit, including qubits sent during preparation). You could just keep repeating the process until it gave you entanglement, then use superdense coding to achieve 2 bits per qubit.

• I see your point, I should have been more specific. Suppose that when the hypothetical protocol fails, it disentangles the Bell pair so that you cannot, on average, use it to violate the Holevo bound. Is it possible to still get lucky and generate this entanglement somehow? Or does the bound apply even in these one-shot instances? From what you say, this hypothetical protocol shouldn't be successful more than 50% of the time since that would certainly violate the bound in the average case. Apr 20, 2022 at 6:30
• @Visipi |00> has 50% overlap with |00> + |11>, so you can just do nothing and "succeed" 50% of the time. Same as you can receive half of any message by flipping coins to generate the bits you receive: you'll get about half the bits right. Apr 20, 2022 at 15:24

No. By definition, the set of states that you can produce under LOCC are the separable states. This includes all possible measurements, post-selection on certain outcomes etc. The whole point of entanglement is that it's the stuff that cannot be made via LOCC. Hence, if you have it, it's a resource that is useful to people who are operating under LOCC restrictions.

• (+1) Off-topic for the question, but could one use LOSR instead and mimic Werner states?
– R.W
Apr 20, 2022 at 7:01
• @R.W Are you meaning "local operations and shared randomness"? It's not a scenario I'm particularly familiar with, but I assume that if you're not interested in a cryptographic setting where there's an eavesdropper, then LOCC is more powerful than LOSR because one user can locally generate randomness and communicate it. (I believe arxiv.org/abs/1106.6095 shows that you cannot generate any entanglement via LOSR) Apr 20, 2022 at 7:15
• Yes, that was exactly what I was referring to. Thanks for the answer and the reference.
– R.W
Apr 20, 2022 at 7:17