Editing Quantum teleportation (section)

== Entanglement swapping ==
{{more|Quantum entanglement swapping}}
Teleportation can be applied not just to pure states, but also [[density matrix|mixed states]], that can be regarded as the state of a single subsystem of an entangled pair. The so-called entanglement swapping is a simple and illustrative example.

If Alice and Bob share an entangled pair, and Bob teleports his particle to Carol, then Alice's particle is now entangled with Carol's particle. This situation can also be viewed symmetrically as follows:

Alice and Bob share an entangled pair, and Bob and Carol share a different entangled pair. Now let Bob perform a projective measurement on his two particles in the Bell basis and communicate the result to Carol. These actions are precisely the teleportation protocol described above with Bob's first particle, the one entangled with Alice's particle, as the state to be teleported. When Carol finishes the protocol she now has a particle with the teleported state, that is an entangled state with Alice's particle. Thus, although Alice and Carol never interacted with each other, their particles are now entangled.

A detailed diagrammatic derivation of entanglement swapping has been given by [[Bob Coecke]],<ref>{{cite report |type=Preprint |last1=Coecke |first1=Bob |title=The logic of entanglement |date=2004 |arxiv=quant-ph/0402014 }}</ref> presented in terms of [[categorical quantum mechanics]].

=== Algorithm for swapping Bell pairs ===
An important application of entanglement swapping is distributing Bell states for use in entanglement distributed [[quantum network]]s. A technical description of the entanglement swapping protocol is given here for pure Bell states.

# Alice and Bob locally prepare known Bell pairs resulting in the initial state: <br/> <math>|\psi\rangle_{\rm in}=|\Phi^+\rangle_{A_1,A_2}|\Phi^+\rangle_{B_1,B_2} </math>
# Alice sends qubit <math>A_1 </math> to a third party Carol
# Bob sends qubit <math>B_1 </math> to Carol
# Carol performs a Bell projection between <math>A_1</math> and <math>B_1</math> that by chance (all four Bell states are possible and recognizable) results in the measurement outcome: <br/> <math>\langle\Phi^+|_{A_1,B_1}|\psi\rangle_{\rm in} = |\Phi^+\rangle_{A_2, B_2}</math>
# In the case of the other three Bell projection outcomes, local corrections given by Pauli operators are made by Alice and or Bob after Carol has communicated the results of the measurement. <br/> <math>\langle\Phi^-|_{A_1,B_1}|\psi\rangle_{\rm in} = \hat{Z}_{B_2}|\Phi^+ \rangle_{A_2,B_2} </math> <br/> <math>\langle\Psi^+|_{A_1,B_1}|\psi\rangle_{\rm in} = \hat{X}_{B_2}|\Phi^+ \rangle_{A_2,B_2} </math> <br/> <math>\langle\Psi^-|_{A_1,B_1}|\psi\rangle_{\rm in} = \hat{X}_{B_2}\hat{Z}_{B_2} |\Phi^+ \rangle_{A_2,B_2} </math>
# Alice and Bob now have a Bell pair between qubits <math>A_2</math> and <math>B_2</math> <br/> <math>|\psi\rangle_{\rm out} = |\Phi^+\rangle_{A_2, B_2} </math>