Understanding entanglement distillation via stabilizer codes?

Question

I am having some trouble understanding the distillation of EPR pairs using a stabilizer code. This idea goes back to the paper by Bennet, DiVincenzo, Smolin, and Wooters.

The idea (I think) is that $k$-EPR pairs can be distilled via an $[n,k]$ stabilizer code $S$. In the protocol two parties share (noisy) $n$-EPR pairs, the code can distill (perfect) $k$-EPR pairs. That is, the decoding and encoding unitaries of the code imply that there is a way in which the parties can teleport a $k$-qubit state meaning they must have (perfect) $k$-EPR pairs. It also seems to be the case that if a correctable error occurs when the $n$-EPR pairs are initially shared it can be corrected.

In particular, the brief Wikipedia article on this subject (which appears to be verbatim from section D. of arxiv.org/pdf/0708.3699) claims that after Alice applies her codespace projection (via measuring the code generators), which via properties of the maximally entangled state applies a similar projection on Bob's side

Alice restores her qubits to the simultaneous +1-eigenspace of the generators in $S$. She sends her measurement results to Bob. Bob measures the generators in $S$. Bob combines his measurements with Alice's to determine a syndrome for the error.

My confusion is understanding the details here. How does Alice "restores her qubits" and how does "Bob combine his measurement with Alice's to determine the error"? Perhaps I am just misunderstanding what's going on in the protocol, but maybe someone here can help me out?

Answer 1

In stabilizer circuits, there's an equivalence between a qubit's worldline just sitting around and a Bell pair preparation or measurement linking two qubits. This is the basis of quantum teleportation, where information is moved through space across a Bell pair (instead of through time by being stored in a qubit).

This equivalence is especially clear in the ZX calculus. For example, in the ZX calculus a Bell pair preparation is a cup and a Bell pair measurement is a cap, so teleportation ends up just looking like the qubit squiggling to the left:

Anyways, the reason I'm talking about this is that, if you take any error correcting code, and replace the timelike worldlines (qubits persisting over time) with spacelike worldlines (Bell prep/measurement linking qubits over space), then distillation is exactly the same thing as normal error correction. Just spread over space instead of over time. All the logic is identical, all the proofs are identical, everything works the same in the light of the equivalence of EPR worldline and qubit worldline. The relationship of the X and Z observables is just across space instead of across time.

The red lines are decoded in exactly the same way as the blue lines. The only difference I can think of is that XX*ZZ = -YY instead of +YY, so stabilizers involving Y terms that would be identical-sign along timelike worldlines can end up opposite-sign along spacelike wordlines; but that's no big deal you just account for the expected sign change. What matters is you know what to expect when there's no errors, not specifically what is expected.