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8.3: Quantum Teleportation at a Glance

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    142682
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    The purpose of this tutorial is to provide a brief mathematical outline of the basic elements of quantum teleportation, as illustrated in the figure below, using matrix and tensor algebra.

    Screen Shot 2019-03-08 at 12.37.41 PM.png

    Alice wishes to teleport the following state (X in the figure) to Bob,

    \[ | \Phi \rangle = a | 0 \rangle + b | 1 \rangle = \begin{pmatrix} a \\ b \end{pmatrix} ~~~ |a|^2 + |b|^2 = 1 \nonumber \]

    where,

    \[ | 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} ~~~ | 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \nonumber \]

    They prepare the following entangled two-particle state, involving A and B, in which Alice has particle A and Bob has B.

    \[ | \Psi_{ab} = \frac{1}{ \sqrt{2}} [|00 \rangle + |11 \rangle] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \nonumber \]

    Alice arranges for the particle to be teleported, |Φ >, and her entangled particle to meet simultaneously on opposite sides of a beam splitter, creating the following the three-particle state.

    \[ | \Phi \rangle | \Psi_{AB} \rangle = \begin{pmatrix} a \\ b \end{pmatrix} \otimes \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} = \frac{1}{ \sqrt{2}} \begin{pmatrix} a \\ 0 \\ 0 \\ a \\ b \\ 0 \\ 0 \\ b \end{pmatrix} \nonumber \]

    This state can be written as a superposition of the following 4-vectors, which are the well-known Bell states. Please see the Appendix for definitions of the Bell states.

    \[ \frac{1}{ \sqrt{2}} \begin{pmatrix} a \\ 0 \\ 0 \\ a \\ b \\ 0 \\ 0 \\ b \end{pmatrix} = \frac{1}{2 \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix} \otimes \begin{pmatrix} a \\ -b \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} b \\ a\ \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} -b \\ a \end{pmatrix} \right] \nonumber \]

    We now write this three-particle state in terms of the Bell basis labels.

    \[ | \Phi \rangle | \Psi_{AB} \rangle = \frac{1}{2} \left[ | \Phi ^+ \begin{pmatrix} a \\ b \end{pmatrix} + | \Phi^- \rangle \begin{pmatrix} a \\ -b \end{pmatrix} + | \Psi^+ \begin{pmatrix} b \\ a \end{pmatrix} + | \Psi^- \begin{pmatrix} -b \\ a \end{pmatrix} \right] \nonumber \]

    Next, Alice makes a Bell-state measurement on her two particles, getting any of the four possible outcomes (Φ+ , Φ- , Ψ+ , or Ψ- ) with equal probability, 25%. Her measurement collapses the state of Bob’s particle into the companion of the result of her Bell-state measurement. Alice then sends the result of her measurement through a classical channel to Bob. Depending on her report, he carries out one of the following operations on his particle to complete the teleportation process.

    \[ | \Phi ^+ \rangle = \hat{I} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix} \nonumber \]

    \[ | \Phi ^- \rangle = \widehat{ \sigma_z} \begin{pmatrix} a \\ -b \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} a \\ -b \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix} \nonumber \]

    \[ | \Psi ^+ \rangle = \widehat{ \sigma_x} \begin{pmatrix} b \\ a \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} b \\ a \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix} \nonumber \]

    \[ | \Psi ^- \rangle = \widehat{ \sigma_z} \widehat{ \sigma_x} \begin{pmatrix} -b \\ a \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} -b \\ a \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix} \nonumber \]

    Appendix

    The Bell basis is the following collection of maximally entangled two-qubit states.

    \[ | \Phi ^+ \rangle = \frac{1}{ \sqrt{2}} [| 00 \rangle + | 11 \rangle] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \nonumber \]

    \[ | \Phi ^- \rangle = \frac{1}{ \sqrt{2}} [| 00 \rangle - | 11 \rangle] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix} \nonumber \]

    \[ | \Psi ^+ \rangle = \frac{1}{ \sqrt{2}} [| 01 \rangle + | 10 \rangle] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} \nonumber \]

    \[ | \Psi ^- \rangle = \frac{1}{ \sqrt{2}} [| 01 \rangle - | 10 \rangle] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} - \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \right] = \frac{1}{ \sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix} \nonumber \]

    The initial entangled state Alice and Bob prepare is the Bell state |Φ+ >.


    This page titled 8.3: Quantum Teleportation at a Glance is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.