A way of transferring quantum information from one place to another. It was proposed by US physicist Charles Bennett and collaborators in 1993, and first demonstrated by Anton Zeilinger and others in 1996. Teleportation involving photons can impose the state of a message photon, for example its spin orientation, on to a distant recipient photon. The state of the original message photon is destroyed. Quantum teleportation exploits the phenomenon of quantum entanglement; it offers reliable movement of quantum information from point to point, for instance in a quantum computer, avoiding problems caused by fragility of quantum states, but it does not imply the ability to move large objects from place to place. In 2004, scientists working independently in the US and Austria performed successful teleportation on atoms by transferring qubits from one atom to another with the help of a third auxiliary atom.
In quantum information, quantum teleportation, or entanglement-assisted teleportation is a technique that transfers a quantum state to an arbitrarily distant location using a distributed entangled state and the transmission of some classical information.
Motivation
This article will use standard nomenclature in quantum information: the two parties are Alice (A) and Bob (B), and a qubit is in general a superposition of quantum state labelled and .
Suppose Alice has a qubit in some arbitrary quantum state . Assume that this quantum state is not known to Alice and she would like to send this state to Bob. Ostensibly, Alice has the following options:
1) She can attempt to physically transport the qubit to Bob.
2) She can broadcast this (quantum) information, and Bob can obtain the information via some suitable receiver. The results of this measurement would be communicated to Bob, who then prepares a qubit in his possession accordingly, to obtain the desired state. (This hypothetical process is called classical teleportation.)
Option 1 is highly undesirable because quantum states are fragile and any perturbation en route would corrupt the state.
The unavailability of option 2 is the statement of the no-broadcast theorem, a consequence of the no cloning theorem. (see reference below.) The parts of a maximally entangled two-qubit state are distributed to Alice and Bob. The protocol then involves Alice and Bob interacting locally with the qubit(s) in their possession and Alice sending two classical bits to Bob.
The result
Suppose Alice has a qubit that she wants to teleport to Bob.
Our quantum teleportation scheme requires Alice and Bob to share a maximally entangled state beforehand, for instance the two-particle Bell state
,or one of the other Bell states. The subscripts A and B in the entangled state refer Alice's or Bob's particle.
So, Alice has two particles (O, the one she wants to teleport, and A, one of the entangled pair), and Bob has one particle, B. In the total system, the state of these three particles is given by
Alice will then make a partial measurement in the Bell basis on the two qubits in her possession. To make the result of her measurement clear, we will rewrite the two qubits of Alice in the Bell basis via the following general identities (these can be easily verified):
and
The three particle state shown above thus becomes:
Notice all we have done so far is a change of basis on Alice's part of the system. Given the above expression, evidently the results of her (local) measurement is that the three-particle state would collapse to one of the following four states (with equal probability of obtaining each):
Alice's two particles are now entangled to each other, in one of the four Bell states. Bob's particle takes on one of the four superposition states shown above. Note how Bob's qubit is now in a state that resembles the state to be teleported. The four possible states for Bob's qubit are unitary images of the state to be teleported.
After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs a unitary operation on his particle to transform it to the desired state :
If Alice indicates her result is , Bob knows his qubit is already in the desired state and does nothing. If the message indicates , Bob would send his qubit through the unitary gate given by the Pauli matrixto recover the state.
Remarks
After this operation, Bob's qubit will take on the state , and Alice's qubit becomes (undefined) part of an entangled state. If we remove the shared entangled state from Alice and Bob, the scheme becomes classical teleportation, which is impossible as mentioned before.Entanglement swapping
Entanglement can be applied not just to pure states, but also mixed states, or even the undefined state of an entangled particle.
If Alice has a particle which is entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.
A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Carol one. Alice's particle and Bob's first particle are entangled, and so are Bob's second and Carol's particle:
___ / \ Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol \___/Now, if Bob performs a projective measurement on his two particles in the Bell state basis and communicates the results to Carol, as per the teleportation scheme described above, the state of Bob's first particle can be teleported to Carol's.
N-state particles
One can imagine how the teleportation scheme given above might be extended to N-state particles, i.e. Bob recovers the desired state by sending his particle through an appropriate unitary gate.
General teleportation scheme
General description
A general teleportation scheme can be described as follows. Systems 2 and 3 are in a maximally entangled state ω that are distributed to Alice and Bob, respectively. The total system is then in the state
A successful teleportation process is a LOCC quantum channel Φ that satisfies
where Tr12 is the partial trace operation with respect systems 1 and 2, and denotes the composition of maps.
Taking adjoint maps in the Heisenberg picture, the success condition becomes
for all observable O on Bob's system. Assume the local measurement have effects
If the measurement registers the i-th outcome, the overall state collapses to
The tensor factor in is while that of is .
Therefore the channel Φ is defined by
Notice Φ satisfies the definition of LOCC. As stated above, the teleportation is said to be successful if, for all observable O on Bob's system, the equality
holds. Assuming all objects are finite dimensional, this becomes
The success criterion for teleportation has the expression
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