Remember Alice and Bob from our previous discussion?
Alice has a muon, which she wants to teleport to Bob. Alice and Bob have made preparations to accomplish this quantum teleportation. Before leaving to go undercover, Alice made sure that a Bell-state analyzer was part of the equipment available to her at "Dark Sciences" laboratory. Then Alice and Bob created pairs of electrons. The process that they used to create the pairs conserves angular momentum, and in that process angular momentum conservation requires that the spin angular momentum of the two electrons adds up to zero. The two electrons are entangled. Alice and Bob put these entangled electrons into individual traps and Alice took one of the electrons from each pair with her to "Dark Sciences", while Bob kept the other one. To teleport the first muon from "Dark Sciences" laboratory to "Globalux" laboratory, Alice now chooses her entangled electron from the first pair and puts it in her Bell-state analyzer, together with the muon.
The Bell-state analyzer makes a measurement whose outcome depends on the spin state of the two-particle system. This measurement can have 4 different outcomes (result 0 – result 3) and after the measurement the spin state of the two-particle system is one of four different states, commonly known as Bell states.
For a muon and an electron the four different Bell states are listed below.
| |y+> = N(|+>m|->e + |->m|+>e) | result 0 |
| |y-> = N(|+>m|->e - |->m|+>e) | result 1 |
| |f+> = N(|+>m|+>e + |->m|->e) | result 2 |
| |f-> = N(|+>m|+>e - |->m|->e) | result 3 |
Alice’s electron has no property of its own, as far as spin is concerned. So even so the other electron of the entangled pair is kept at a far away location, it must be included in the description of the spin state before the measurement. The notation for this spin-0 entangled state of the two electrons is
N(|+>eA|->eB - |->eA|+>eB),
with the normalization constant N = 2–½. The subscripts A and B refer to the owners of the electron, Alice and Bob.
The muon is in the unknown state (a|+>m + b|->m), so the state of the electron-muon system in the Bell-state analyzer before the measurement is
(a|+>m
+ b|->m)
N(|+>eA|->eB - |->eA|+>eB)
= N(a|+>m|+>eA|->eB
- a|+>m|->eA|+>eB
+ b|->m|+>eA|->
eB - b|->m|->eA|+>eB).
The rightmost electron belongs to Bob.
We can rewrite this expression in terms of the Bell states.
(Click
here if you are interested in the algebraic details.)
N (a|+>m|+>eA|->eB
- a|+>m|->eA|+>eB
+ b|->m|+>eA|->eB
- b|->m|->eA|+>eB)
=
- ½|y+>(a|+>eB
- b|->eB)
- ½|y->(a|+>eB
+ b|->eB)
+ ½|f+>(a|->eB
- b|+>eB)
+ ½|f->(a|->eB
+ b|+>eB)
When Alice makes a measurement with the Bell state analyzer, every outcome (|y+>, |y->, |f+>, |f->) is equally likely. Each outcome has probability (½)2 = ¼. Once Alice has made a measurement, Bob’s electron has been put into a particular state. Alice notes the result of the Bell-state analyzer and transmits the result number (00 or 01 or 10 or 11 in binary form) to Bob at approximately the speed of light, using a radio transmission, a satellite phone, etc. When Bob receives the result of Alice's measurement over a classical channel, he knows which of the states listed below is the spin state of his electron, even though he does not know the coefficients a and b.
| Result: | Alice's Bell state: | Bob's electron state: |
| 0 | |y+> | a|+>eB - b|->eB |
| 1 | |y-> | a|+>eB + b|->eB |
| 2 | |f+> | a|->eB - b|+>eB |
| 3 | |f-> | a|->eB + b|+>eB |
Given this result Bob knows exactly what to do to put his electron into the original muon spin state, and then transfer this state to one of his muons. Alice's muon has been teleported.
Procedure:
Bob can use phase shift, bit flip, and CNOT operations to produce a muon in the original spin state a|+> + b|-> after he receives the result of Alice's measurement. The appropriate operations are listed in the table below.
| Alice' result: | Bob's electron state: | Sequence of operations to put the electron in the state a|+>eB + b|->eB: |
Sequence of operations to transfer this state to a muon originally in the |+>m state: |
| 0 | a|+>eB - b|->eB | Phase shift | |
| 1 | a|+>eB + b|->eB | Identity (do nothing) | |
| 2 | a|->eB - b|+>eB | Phase shift followed by bit flip | |
| 3 | a|->eB + b|+>eB | Bit flip |
Summary:
The result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. This is known as "non-local behavior".
Quantum teleportation using the above outlined scheme was accomplished for the first time in 1997. The quantum state of a photon was teleported. This website steps you through the experiment.
Other Links: