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Quantum Teleportation: Difference between revisions
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* 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 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. | * 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 <math>|\psi\rangle</math> 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:</br> | |||
<math>|\psi\rangle = \alpha |0\rangle_{O} + \beta |1\rangle_{O}</math>, <math>\alpha</math> and <math>\beta</math> coefficients are unknown to the Sender.</br> | |||
'''<u>Stage</u>''' Share entangled qubits (EPR pair) | |||
# 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:</br> | |||
<math>|\Phi^+\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B)</math> | |||
#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: | |||
<math>|\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)</math></br> | |||
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 <math>|\Phi^+\rangle$, $|\Phi^-\rangle$, $|\Psi^+\rangle$ and $|\Psi^-\rangle</math>)</br></br> | |||
<math>|\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]</math></br></br> | |||
'''<u>Stage 2</u>''' Local Measurement by Sender(A) | |||
* '''Input:''' <math>|\psi\rangle_O \otimes |\Phi^+\rangle_{AB}</math> | |||
* '''Output:''' The output of the Sender's measurement in Bell basis | |||
# 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. | |||
# The output of this measurement will be one of the four Bell states: $|\Phi^+\rangle$, $|\Phi^-\rangle$, $|\Psi^+\rangle$ and <math>|\Psi^-\rangle</math></br></br> | |||
'''<u>Stage 3</u>''' Send classical information | |||
# 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: | |||
## '''if''' the result is <math>|\Phi^+\rangle \rightarrow</math> send <math>00</math> | |||
##'''if''' the result is <math>|\Phi^-\rangle \rightarrow</math> send <math>01</math> | |||
## '''if''' the result is <math>|\Psi^+\rangle \rightarrow</math> send <math>10</math> | |||
## '''if''' the result is <math>|\Psi^-\rangle \rightarrow</math> send <math>11</math></br></br> | |||
'''<u>Stage 4</u>''' Local Operation by Receiver(B) | |||
*'''Input:''' two classical bits: c <math>\in \{00, 01, 10, 11\}</math> | |||
*'''Output:''' Teleported state <math>|\psi\rangle</math> | |||
# 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: | |||
# '''if''' the result is <math>|\Phi^+\rangle \rightarrow</math> Receiver's state will be: <math>\alpha |0\rangle + \beta |1\rangle</math> | |||
# '''if''' the result is <math>|\Phi^-\rangle \rightarrow</math> Receiver's state will be: <math>\alpha |0\rangle - \beta |1\rangle</math> | |||
# '''if''' the result is <math>|\Psi^+\rangle \rightarrow</math> Receiver's state will be: <math>\beta |0\rangle + \alpha |1\rangle</math> | |||
# '''if''' the result is <math>|\Psi^-\rangle \rightarrow</math> Receiver's state will be: <math>\beta |0\rangle - \alpha |1\rangle</math> | |||
* The receiver will perform following Operators on the above states: | |||
# '''if''' he receives <math>00 \rightarrow</math> he performs <math>I</math> (does nothing) | |||
# '''if''' he receives <math>01 \rightarrow</math> he performs <math>Z([[Pauli Z]])</math> | |||
# '''if''' he receives <math>10 \rightarrow</math> he performs <math>X([[Pauli X]])</math> | |||
# '''if''' he receives <math>11 \rightarrow</math> he performs <math>ZX</math> (Pauli X then a Pauli Z) | |||
*As a result, the state of the receiver will be: <math>|\psi\rangle_B = \alpha|0\rangle + \beta |1\rangle</math> | |||
==Further Information== | |||