Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle (or series of particles) to another particle (or series of particles) in another location via quantum entanglement. It does not transport energy or matter, nor does it allow communication of information at superluminal (faster than light) speed. Its distinguishing feature is that it can transmit the information present in a quantum superposition, useful for quantum communication and computation.

Motivation

The two parties are Alice (A) and Bob (B), and a qubit is, in general, a superposition of quantum state labeled |0\rangle and |1\rangle. Equivalently, a qubit is a unit vector in two-dimensional Hilbert space.

Suppose Alice has a qubit in some arbitrary quantum state |\psi\rangle. 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.

3. She can perhaps measure the unknown qubit in her possession. 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.

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.

Assume that Alice and Bob share an entangled qubit AB. That is, Alice has one half, A, and Bob has the other half, B. Let C denote the qubit Alice wishes to transmit to Bob.

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.

Remarks

* After this operation, Bob's qubit will take on the state |\psi\rangle= \alpha |0\rangle + \beta|1\rangle, and Alice's qubit becomes (undefined) part of an entangled state. Teleportation does not result in the copying of qubits, and hence is consistent with the no cloning theorem.

* There is no transfer of matter or energy involved. Alice's particle has not been physically moved to Bob; only its state has been transferred. The term "teleportation", coined by Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters, reflects the indistinguishability of quantum mechanical particles.

* The teleportation scheme combines the resources of two separately impossible procedures. If we remove the shared entangled state from Alice and Bob, the scheme becomes classical teleportation, which is impossible as mentioned before. On the other hand, if the classical channel is removed, then it becomes an attempt to achieve superluminal communication, again impossible (see no communication theorem).

* For every qubit teleported, Alice needs to send Bob two classical bits of information. These two classical bits do not carry complete information about the qubit being teleported. If an eavesdropper intercepts the two bits, she may know exactly what Bob needs to do in order to recover the desired state. However, this information is useless if she cannot interact with the entangled particle in Bob's possession.

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