Teleportation Protocol

The objective of this technique is to transmit one [[Qubit|qubit]] between Alice and Bob by sending two classical bits. However, Alice and Bob must initially share one [[Entanglement|entangled state]]. Alice and Bob perform the following steps: * Alice and Bob initially share a [[Bell State]]

\beta_{00} = \frac{1}{\sqrt{2}}({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})

. \beta_{00} . * Alice applies [[Hadamard |Hadamard]]to the first of her two qubits, then performs a [[Bell measurement|Bell measurement]] on both and sends the result of the measurement (2 classical bits) to Bob. --> * Alice performs a joint measurement upon the qubit she wants to transmit and her half of the [[Bell State]]. This measurement may be performed by applying a [[CNOT|CNOT]] between the qubit to be transmitted and the first one on the pair

\beta_{00}

and then applying [[Hadamard |Hadamard]] to the first of her two qubits. Measuring both the qubits in the computational basis completes the Bell measurement. * Alice then sends the results of the measurements on her two qubits (2 classical bits

b_1,b_2

) to Bob. * Bob applies a transformation upon his qubit, according to the two received bits, based on the following table Received bits Gate to be applied

00

I

01

X

10

Z

11

ZX

The complete circuit is the following [[Image:telep.jpg|center|500px]] where

{\left\vert{\psi}\right\rangle}

is the qubit to be teleported. Let's see, let

{\left\vert{\psi}\right\rangle}=\alpha{\left\vert{0}\right\rangle}+\beta{\left\vert{1}\right\rangle}

then

{\left\vert{\psi}\right\rangle} \otimes \beta_{00}

=(\alpha{\left\vert{0}\right\rangle}+\beta{\left\vert{1}\right\rangle})\left(\frac{1}{\sqrt{2}}({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})\right)

=\frac{1}{\sqrt{2}}\left(\alpha{\left\vert{0}\right\rangle}({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})+\beta{\left\vert{1}\right\rangle}({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})\right)

{\;{{CNOT(1,2)} \atop \longrightarrow}\;}
\frac{1}{\sqrt{2}}\left(\alpha{\left\vert{0}\right\rangle}({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})+\beta{\left\vert{1}\right\rangle}({\left\vert{10}\right\rangle}+{\left\vert{01}\right\rangle})\right)

{\;{{H(1)} \atop \longrightarrow}\;}
\frac{1}{\sqrt{2}}\left(\alpha\frac{1}{\sqrt{2}}({\left\vert{0}\right\rangle}+{\left\vert{1}\right\rangle})({\left\vert{00}\right\rangle}+{\left\vert{11}\right\rangle})+\beta\frac{1}{\sqrt{2}}({\left\vert{0}\right\rangle}-{\left\vert{1}\right\rangle})({\left\vert{10}\right\rangle}+{\left\vert{01}\right\rangle})\right)

=\frac{1}{2}\left({\left\vert{00}\right\rangle}(\alpha{\left\vert{0}\right\rangle}+\beta{\left\vert{1}\right\rangle}) +{\left\vert{01}\right\rangle}(\alpha{\left\vert{1}\right\rangle}+\beta{\left\vert{0}\right\rangle})
+{\left\vert{10}\right\rangle}(\alpha{\left\vert{0}\right\rangle}-\beta{\left\vert{1}\right\rangle})
+{\left\vert{11}\right\rangle}(\alpha{\left\vert{1}\right\rangle}-\beta{\left\vert{0}\right\rangle})\right)

=\frac{1}{2}\sum_{b_1b_2=0}^{1}{\left\vert{b_1 b_2}\right\rangle}(X^{b_2}Z^{b_1}){\left\vert{\psi}\right\rangle}

Therefore, appliyng

Z^{b_1}X^{b_2}

Bob will obtain the original state

{\left\vert{\psi}\right\rangle}

. == References == *Extracted and translated from the following text: [http://www.fceia.unr.edu.ar/~diazcaro/QC/Tutorials/Charlas%20Introductorias%20a%20la%20Computacion%20Cuantica.ps Charlas Introductorias a la Computación Cuántica] by [[User:Janus|Alejandro Díaz-Caro]] *[http://www.cco.caltech.edu/~qoptics/teleport.html Caltech Teleportation Experiment] from [[Kimble Lab]] *[http://www.research.ibm.com/quantuminfo/teleportation/index.html Teleportation page] from [[IBM Quantum Group]] [[Category:Introductory Tutorials]] [[Category:Handbook of Quantum Information]]

Teleportation Protocol

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