Editing Quantum Teleportation (section)

==Protocol Description==

[https://github.com/quantumprotocolzoo/protocols/tree/master/QuantumStateTeleportation <u>click here for Python code</u>]

[https://github.com/apassenger/CQC-Python/tree/master/examples/pythonLib/teleport/Quantum%20State%20Teleportation <u>click here for SimulaQron code</u>]

*'''Input:''' The qubit <math>|\psi\rangle</math> is the to-be-sent state which the first party (the sender) wants to transfer to the second party (the receiver). The quantum state can be written generally in standard basis as:
<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></br>
'''<u>Stage 1</u>''' Share entangled qubits (EPR pair)</br>
#  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</math>, <math>|\Phi^-\rangle</math>, <math>|\Psi^+\rangle</math> and <math>|\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 the 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: <math>|\Phi^+\rangle</math>, <math>|\Phi^-\rangle</math>, <math>|\Psi^+\rangle</math> 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, the 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 the 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</math> ([[Pauli Z]])
# '''if'''  he receives <math>10 \rightarrow</math>, he performs <math>X</math> ([[Pauli X]])
# '''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>