Quantum Teleportation

Quantum teleportation is a protocol by which a quantum state (or information stored in a quantum state) can be transmitted physically from one location (or one party) to another. This example protocol requires sharing an entangled state like an EPR pair between two parties and also allowing the parties to communicate classically (sending bits of information). Quantum Teleportation can be treated as a send/receive scheme for qubits. Quantum teleportation provides a mechanism of sending an unknown qubit from one location to another, without physically moving the particle. This task can be done due to the existence of long-range correlations between entangled pairs. The quantum teleportation is used widely as a basic protocol in many other quantum communication and quantum cryptography protocols.

Tags: Category: Building Blocks, teleportation, quantum communication, sending quantum information, send/receive in the quantum network, Category: Quantum Functionality, Category: Specific Task

Assumptions

• The protocol is deterministic i.e. the entangled state and the measurements and gates are assumed perfect, the protocol will always succeed
• During the protocol, value of $\alpha$ and $\beta$ will remain unknown to both the parties (and any adversary as well)
• A public classical channel is assumed between the two parties
• There is no transfer of matter or energy involved. Sender's particle has not been physically moved to Receiver; only the particle's state has been transferred

Outline

The quantum teleportation protocol begins with a quantum state or qubit, in the possession of the first party (The sender). We need this quantum state to be transferred to the second party (The receiver). This state is unknown to both parties meaning that the Sender does not know the representation of the qubit on any basis. Before starting the protocol the two parties must share an entangled state (for example an EPR pair). The entangled state here is a two-qubit state where each party has one share of these qubits which have a special quantum correlation. After sharing the entangled state, the parties can take an arbitrary distance (In theory, without any noise and by assuming that the entanglement can be held for an arbitrary distance which is not the case in the real experiments). After this preparation stage, the two parties will perform the protocol as follows:

• At Sender's location, a Bell measurement of the EPR pair qubit and the qubit to be teleported is performed, yielding one of four measurement outcomes, which can be encoded in two classical bits of information. Both qubits at Sender's location are then discarded.
• Using the classical channel, the two bits are sent from Sender to Receiver.
• As a result of the measurement performed at Sender's location, the EPR pair qubit at Receiver's location is in one of four possible states. Of these four possible states, one is identical to the original quantum state, and the other three are closely related. Which of these four possibilities actually obtained, is encoded in the two classical bits. Knowing this, the EPR pair qubit at Receiver's location is modified by local unitary operations that the Receiver performs on his state. And the result will be the original qubit.

Notations

• $|\psi \rangle _{O}:$ The unknown original state to be teleported from Sender to Receiver
• $|\Phi ^{+}\rangle _{AB}:$ The EPR pair (or Bell state) shared between two parties
• Bell States($|\Phi ^{+}\rangle$, $|\Phi ^{-}\rangle$, $|\Psi ^{+}\rangle$ and $|\Psi ^{-}\rangle$): Set of orthonormal two-qubit states having the maximum amount of entanglement. These states can be used as a basis for a two-qubit quantum system.
• $I:$ The identity operator
• $X,Y,Z:$ The Pauli Operators