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

Properties

• This protocol uses a public classical channel to transfer two bits of classical information.
• The teleportation protocol uses entanglement (or entangled EPR states) as a resource.
• The teleportation protocol is secure against cloning attacks, as a result of no-cloning theorem in quantum mechanics i.e. any of the involved states in the protocol cannot perfectly be copied. Also, any other interference will affect the shared state between the two parties and the attack will be discovered.
• The teleportation protocol is secure against entanglement attacks because of the monogomy of entanglement in quantum mechanics. It means that if an adversary tries to entangle her state with the shared EPR pair, the amount of the entanglement of the shared state between two parties will change and the attacker will be discovered.
• The size of the classical information sent by the Sender to the Receiver is infinitely smaller than the information required to give a classical description of the teleported quantum state.

Pseudo Code

• Input: The qubit $|\psi \rangle$ is the to-be-send state which the first party(Sender) wants to transfer to the second party (Receiver). The quantum state can be written generally in standard basis as:

$|\psi \rangle =\alpha |0\rangle _{O}+\beta |1\rangle _{O}$, $\alpha$ and $\beta$ coefficients are unknown to the Sender.

Stage 1 Share entangled qubits (EPR pair)

1. Generate an EPR pair (or a maximally-entangled two-qubit sate) and give one qubit to the sender(A) and one to the receiver(B). The shared EPR state between the two parties is described as:
   

$|\Phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}|0\rangle _{B}+|1\rangle _{A}|1\rangle _{B})$

1. This step is a pre-preparation step which should be run before the protocol starts. The state of all the three particles are as follows:

$|\psi \rangle _{O}\otimes |\Phi ^{+}\rangle _{AB}=(\alpha |0\rangle _{O}+\beta |1\rangle _{O})\otimes {\frac {1}{\sqrt {2}}}(|0\rangle _{A}|0\rangle _{B}+|1\rangle _{A}|1\rangle _{B})$
In order to make the next step more clear, the above three-qubit states can be written in Bell basis (spaned by four two-qubit Bell states $|\Phi ^{+}\rangle$, $|\Phi ^{-}\rangle$, $|\Psi ^{+}\rangle$ and $|\Psi ^{-}\rangle$)

$|\psi \rangle _{O}\otimes |\Phi ^{+}\rangle _{AB}={\frac {1}{2}}[|\Phi ^{+}\rangle _{AO}\otimes (\alpha |0\rangle +\beta |1\rangle )_{B}+|\Phi ^{-}\rangle _{AO}\otimes (\alpha |0\rangle -\beta |1\rangle )_{B}+|\Psi ^{+}\rangle _{AO}\otimes (\beta |0\rangle +\alpha |1\rangle )_{B}+|\Psi ^{-}\rangle _{AO}\otimes (\beta |0\rangle -\alpha |1\rangle )_{B}]$

Stage 2 Local Measurement by Sender(A)

• Input: $|\psi \rangle _{O}\otimes |\Phi ^{+}\rangle _{AB}$
• Output: The output of the Sender's measurement in Bell basis

1. The sender(A) performs a local measurement on two qubits that she has (the original state and her share of the EPR pair) in the Bell basis.
2. The output of this measurement will be one of the four Bell states: $|\Phi^+\rangle$, $|\Phi^-\rangle$, $|\Psi^+\rangle$ and $|\Psi ^{-}\rangle$
   
   

Stage 3 Send classical information

1. According to the result of the measurement on the previous step, Sender A sends two bits of classical information to B indicating the result of her measurement:
   1. if the result is $|\Phi ^{+}\rangle \rightarrow$ send $00$
   2. if the result is $|\Phi ^{-}\rangle \rightarrow$ send $01$
   3. if the result is $|\Psi ^{+}\rangle \rightarrow$ send $10$
   4. if the result is $|\Psi ^{-}\rangle \rightarrow$ send $11$
      
      

Stage 4 Local Operation by Receiver(B)

• Input: two classical bits: c $\in \{00,01,10,11\}$
• Output: Teleported state $|\psi \rangle$

1. The receiver performs a local unitary operation on his qubit. Before this step and after that the two-qubit measurement is performed by the sender, The state of the Receiver will change to the following states according to the Sender's measurement results:
2. if the result is $|\Phi ^{+}\rangle \rightarrow$ Receiver's state will be: $\alpha |0\rangle +\beta |1\rangle$
3. if the result is $|\Phi ^{-}\rangle \rightarrow$ Receiver's state will be: $\alpha |0\rangle -\beta |1\rangle$
4. if the result is $|\Psi ^{+}\rangle \rightarrow$ Receiver's state will be: $\beta |0\rangle +\alpha |1\rangle$
5. if the result is $|\Psi ^{-}\rangle \rightarrow$ Receiver's state will be: $\beta |0\rangle -\alpha |1\rangle$

• The receiver will perform following Operators on the above states:

1. if he receives $00\rightarrow$ he performs $I$ (does nothing)
2. if he receives $01\rightarrow$ he performs $Z$(Pauli Z)
3. if he receives $10\rightarrow$ he performs $X$(Pauli X)
4. if he receives $11\rightarrow$ he performs $ZX$ (Pauli X then a Pauli Z)

• As a result, the state of the receiver will be: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle_B = \alpha|0\rangle + \beta |1\rangle}