8.9: A Quantum Teleportation Experiment for Undergraduates

This Mathcad document examines the math involved in a teleportation experiment for undergraduates using IBM's 5-qubit quantum processor (IBM Quantum Experience) posted by S. Fedortchenko at arXiv:1607.02398v1. Except for state preparation it is identical to the following teleportation circuit, which can be found in my "Teleportation: Another Look."

\[ \begin{matrix} ~ & \text{Initial} & ~ & ~ & ~ & ~ & ~ & \text{Final} \\ ~ & | \Psi \rangle & \cdots & \cdot & \cdots & \fbox{H} & \triangleright & \text{Measure}~ | \text{a} \rangle~ \text{0 or 1} \\ \text{Alice} & ~ & ~ & | & ~ & ~ & ~ & \text{Bell state measurement} \\ ~ & \cdot & \cdots & \oplus & \cdots & \cdots & \triangleright & \text{Measure}~ | \text{b} \rangle~ \text{0 or 1} \\ ~ & \beta_{00} \\ \text{Bob} & \cdot & \cdots & \cdots & \cdots & \cdots & \triangleright & X^b Z^a \rightarrow | \Psi \rangle \end{matrix} \nonumber \]

Fedortchenko's teleportation circuit is shown below.

\[ \begin{matrix} \text{Initial} & ~ & 1 & 2 & 3 & 4 & 6 & \text{Final} \\ |0 \rangle & \triangleright & \fbox{H} & \fbox{T} & \fbox{H} & \fbox{S} & \cdot & \fbox{H} & \triangleright & \text{Measure}~ | \text{a} \rangle~ \text{0 or 1} \\ ~&~&~&~&~&~ & | & ~ & ~ & \text{Bell state measurement} \\ |0 \rangle & \triangleright & \cdots & \cdots & \cdots & \oplus & \oplus & \cdots & \triangleright & \text{Measure}~ | \text{b} \rangle~ \text{0 or 1} \\ ~&~&~&~&~ & | \\ |0 \rangle & \triangleright & \cdots & \cdots & \fbox{H} & \cdot & \cdots & \cdots & \triangleright & X^b Z^a \rightarrow | \Psi \rangle \end{matrix} \nonumber \]

Single qubit operators:

\[ \begin{matrix} I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} & H = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} & S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \\ T = \begin{pmatrix} 1& 0 \\ 0 & e^{ i \frac{ \pi}{4}} \end{pmatrix} & X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} & Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{matrix} \nonumber \]

Two qubit operators:

\[ \begin{matrix} \text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} & \text{ICNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \end{matrix} \nonumber \]

Demonstrate the generation of the teleportee state held by Alice:

\[ \text{SHTH} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

\[ e^{i \frac{ \pi}{8}} \left[ \cos \left( \frac{ \pi}{8} \right) \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \sin \left( \frac{ \pi}{8} \right) \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

Demonstrate the creation of the entangled Bell state shared by Alice and Bob:

\[ \begin{matrix} \text{ICNOT (kronecker(I, H))} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.707 \\ 0 \\ 0 \\ 0.707 \end{pmatrix} & \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] \end{matrix} \nonumber \]

Creation of the teleportee state and the entangled Bell state occurs in the first four steps.

\[ \text{StatePrep = kronecker (S, ICNOT) kronecker (H, kronecker(I, H)) kronecker (T, kronecker(I, I)) kronecker (H kronecker (I, I))} \nonumber \]

Teleportation occurs in steps 5 and 6.

\[ text{TC = kronecker (H, kronecker (I, I)) kronecker (CNOT, I)} \nonumber \]

\[ \text{TC StatePrep} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.427 + 0.177i \\ 0.177 + 0.073i \\ 0.177 + 0.073i \\ 0.427 + 0.177i \\ 0.427 + 0.177i \\ -0.177 - 0.073i \\ -0.177 - 0.073i \\ 0.427 + 0.177i \end{pmatrix} \nonumber \]

Measurement occurs in the final step on the top two wires with possible outcomes |00>, |01>, |10> and |11>. In other words, Alice makes a Bell state measurement (see first figure above) on the two qubits in her possession and informs Bob of the result through a classical channel. He then performs an operation on his qubit to recover the teleported state.

If Alice observes |00> Bob does nothing (the identity operation) because he has the teleportee on his register. If Alice observes |01> Bob applies the X operator, if she finds |10> he uses the Z operator, and finally if Alice observes |11> Bob applies the X operator followed by the Z operator. Further mathematical detail is provided by showing explicitly the four equally probable measurement outcomes that Alice observes, and Bob's subsequent action on his register.

Computational Details

Measurement operator for |0>:

\[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \nonumber \]

Measurement operator for |1>:

\[ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \nonumber \]

Alice measures |00>:

\[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

Bob's action:

\[ I \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

\[ 2 \text{kronecker} \left[ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \text{kronecker} \left[ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \text{I} \right] \right] \text{TC StatePrep} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \nonumber \]

Alice measures |01>:

\[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0.354 + 0.146i \\ 0.854 + 0.354i \end{pmatrix} \nonumber \]

Bob's action:

\[ X \begin{pmatrix} 0.354 + 0.146i \\ 0.854 + 0.354i \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

\[ 2 \text{kronecker} \left[ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \text{kronecker} \left[ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \text{I} \right] \right] \text{TC StatePrep} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0.354 + 0.146i \\ 0.854 + 0.354i \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \nonumber \]

Alice measures |10>:

\[ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ -0.354 - 0.146i \end{pmatrix} \nonumber \]

Bob's action:

\[ Z \begin{pmatrix} 0.854 + 0.354i \\ -0.354 - 0.146i \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

\[ 2 \text{kronecker} \left[ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \text{kronecker} \left[ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \text{I} \right] \right] \text{TC StatePrep} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0.854 + 0.354i \\ -0.354 - 0.146i \\ 0 \\ 0 \end{pmatrix} \nonumber \]

Alice measures |11>:

\[ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -0.354 - 0.146i \\ 0.854 + 0.354i \end{pmatrix} \nonumber \]

Bob's action:

\[ Z X \begin{pmatrix} 0.854 + 0.354i \\ -0.354 - 0.146i \end{pmatrix} = \begin{pmatrix} 0.854 + 0.354i \\ 0.354 + 0.146i \end{pmatrix} \nonumber \]

\[ 2 \text{kronecker} \left[ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \text{kronecker} \left[ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \text{I} \right] \right] \text{TC StatePrep} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ -0.354 - 0.146i \\ 0.854 + 0.354i \end{pmatrix} \nonumber \]