More precisely, quantum teleportation is a quantum protocol by which a qubit a (the basic unit of quantum information) can be transmitted exactly (in principle) from one location to another. The prerequisites are a conventional communication channel capable of transmitting two classical bits (i.e. one of four states), and an entangled pair (b,c) of qubits, with b at the origin and c at the destination. (So whereas b and c are intimately related, a is entirely independent of them other than being initially colocated with b.) The protocol has three steps: measure a and b jointly to yield two classical bits; transmit the two bits to the other end of the channel (the only potentially time-consuming step, due to speed-of-light considerations); and use the two bits to select one of four ways of recovering c. The upshot of this protocol is to permute the original arrangement ((a,b),c) to ((b′,c′),a), that is, a moves to where c was and the previously separated qubits of the Bell pair turn into a new Bell pair (b′,c′) at the origin.
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:
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.
Similarly, it has also been shown formally that classical teleportation, aka. option 3, is impossible; this is called the no teleportation theorem. This is another way to say that quantum information cannot be measured reliably.
Thus, Alice seems to face an impossible problem. A solution was discovered by Bennet et al. (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. In the end, the qubit in Bob's possession will be in the desired state.
Alice applies a unitary operation on the qubits AC and measures the result to obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, B, now contains information about C; however, the information is somewhat randomized. More specifically, Bob's qubit B is in one of four states uniformly chosen at random and Bob cannot obtain any information about C from his qubit.
Alice provides her two measured qubits, which indicate which of the four states Bob possesses. Bob applies a unitary transformation which depends on the qubits he obtains from Alice, transforming his qubit into an identical copy of the qubit C.
Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally as:
Alice takes one of the particles in the pair, and Bob keeps the other one. The subscripts A and B in the entangled state refer to Alice's or Bob's particle. We will assume that Alice and Bob share the entangled state .
So, Alice has two particles (C, 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):
The three particle state shown above thus becomes the following four-term superposition:
Notice all we have done so far is a change of basis on Alice's part of the system. No operation has been performed and the three particles are still in the same state. The actual teleportation starts when Alice measures her two qubits in the Bell basis. 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. The entanglement originally shared between Alice's and Bob's is now broken. 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.
The crucial step, the local measurement done by Alice on the Bell basis, is done. It is clear how to proceed further. Alice now has complete knowledge of the state of the three particles; the result of her Bell measurement tells her which of the four states the system is in. She simply has to send her results to Bob through a classical channel. Two classical bits can communicate which of the four results she obtained.
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 :
to recover the state.
to his qubit.
Teleportation is therefore achieved.
Experimentally, the projective measurement done by Alice may be achieved via a series of laser pulses directed at the two particles.
In the literature, one might find alternative, but completely equivalent, descriptions of the teleportation protocol given above. Namely, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can also be implemented by quantum gates. Direct calculation shows that this gate is given by
Entanglement can be applied not just to pure states, but also mixed states, or even the undefined state of an entangled particle. The so-called entanglement swapping is a simple and illustrative example.
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. Although Alice and Carol never interacted with each other, their particles are now entangled.
One can imagine how the teleportation scheme given above might be extended to N-state particles, i.e. particles whose states lie in the N dimensional Hilbert space. The combined system of the three particles now has a dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the dimensional subsystem. This measurement has equally probable outcomes, which are then communicated to Bob classically. Bob recovers the desired state by sending his particle through an appropriate unitary gate.
A general teleportation scheme can be described as follows. Three quantum systems are involved. System 1 is the (unknown) state ρ to be teleported by Alice. 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
where Tr12 is the partial trace operation with respect systems 1 and 2, and denotes the composition of maps. This describes the channel in the Schrödinger picture.
Taking adjoint maps in the Heisenberg picture, the success condition becomes
for all observable O on Bob's system. The tensor factor in is while that of is .
The proposed channel Φ can be described more explicitly. To begin teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in her possession. 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 . Bob then applies a corresponding local operation Ψi on system 3. On the combined system, this is described by
where Id is the identity map on the composite system .
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. The left hand side of the equation is:
where Ψi* is the adjoint of Ψi in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes
The success criterion for teleportation has the expression