LOCC

Ch. 14's teleportation protocol used exactly two ingredients beyond the shared Bell pair: things Alice and Bob each did locally to their own qubit, and an ordinary phone call. That combination has a name — Local Operations and Classical Communication, LOCC — and it's the formal definition of "everything you're allowed to do without directly sending quantum information."

What counts as LOCC

• Local operations: any gate or measurement Alice applies to her own qubits, and any gate or measurement Bob applies to his — never anything that touches both at once.
• Classical communication: sending ordinary bits back and forth, as many rounds as needed, including letting Bob's next move depend on what Alice tells him.

What's explicitly not allowed: any gate (like CNOT) that acts jointly on a qubit of Alice's and a qubit of Bob's. That single restriction turns out to be enormously consequential.

The central fact: LOCC cannot create entanglement

If Alice and Bob start with a product state — no entanglement between them at all — no sequence of local operations and classical messages, however clever, can ever produce an entangled state. Try to break this yourself:

LOCC can't create entanglement, but it can move it around

Here's the part that surprises people: LOCC genuinely cannot increase the total entanglement in a system — but it absolutely can redistribute entanglement that's already there, sometimes in startling ways. Entanglement swapping is the showcase example: start with two completely independent Bell pairs, A-B and C-D (A,D never having touched each other, ever). Perform a joint (non-local) Bell measurement on just B and C, then tell A and D's owners the outcome over a classical channel. The result: A and D end up in a Bell state — perfectly entangled with each other, despite never interacting.

This isn't a violation of "LOCC can't create entanglement" — the entanglement in A-D was always implicitly present in the combined 4-particle system; the swap just relocated which two particles carry it. The total amount of entanglement in the full system never went up.

Why "LOCC-equivalence" is the right notion of "same entanglement"

Because LOCC can never increase entanglement, it gives quantum information theory exactly the right yardstick: two states are considered to have "the same kind" of entanglement if and only if LOCC can convert each into the other (with some probability, in either direction). This is precisely the distinction Ch. 17 leans on: GHZ and W states, for 3+ qubits, turn out to be LOCC-inequivalent — neither can be converted into the other by any sequence of local operations and classical communication, which is the rigorous version of "they're genuinely different kinds of entanglement," not just different-looking formulas.