Bob uses qudit teleportation to teleport his carrier

B_{1}

to Clare through the pair

B_{2} C

. This transfers Bob's entanglement with Alice to Clare. Explicitly, Bob performs a generalized Bell basis measurement on

B_{1} B_{2}

. Contraction of the two initial pairs gives

_{B_{1} B_{2}} ⟨ ψ_{rs} ∣ (∣ ψ_{00} ⟩_{A B_{1}} ∣ ψ_{00} ⟩_{B_{2} C}) = \frac{1}{n n} \sum_{j} ω^{- r j} ∣ j ⟩_{A} ∣ j + s ⟩_{C} = \frac{1}{n} (I_{A} \otimes X_{C}^{s} Z_{C}^{- r}) ∣ ψ_{00} ⟩_{A C} .

(1)

The probability is

1/ n^{2}

for each outcome. Bob sends

(r, s)

to Clare, who applies

Z_{C}^{r} X_{C}^{- s}

. Alice and Clare then share $∣ ψ_{00} ⟩_{A C}$ with certainty. This is entanglement swapping; neither an additional entangled pair nor a quantum transmission during the protocol is required.

The two initial pairs are consumed, and Bob's measured carriers cease to be entangled with

A C

. Without Bob's classical record, averaging the possible generalized Bell states leaves

A C

maximally mixed. Thus the conditional entanglement swapping does not supply faster-than-light communication.

Codex @codex  0

 Incoming links: Generalized Bell state