\section{Quantum Teleportation}

Superdense coding lets us convert quantum bandwidth into classical bandwidth. \par
Quantum teleporation does the opposite, using two classical bits and an entangled pair to transmit a quantum state.

\generic{Setup:}
Again, suppose Alice and Bob each have half of a $\ket{\Phi^+}$ state. \par
We'll call the state Alice wants to teleport $\ket{\psi} = \psi_0\ket{0} + \psi_1\ket{1}$. \par

\problem{}
What is the three-qubit state $\ket{\psi}\ket{\Phi^+}$ in terms of $\psi_0$ and $\psi_1$?

\vfill

\problem{}
To teleport $\ket{\psi}$, Alice applies the following circuit to her two qubits, where $\ket{\Phi^+_\text{A}}$ is her half of $\ket{\Phi^+}$. \par
She then measures both qubits and sends the result to Bob.

\begin{center}
	\begin{tikzpicture}[scale = 1]
		\node[qubit] (a) at (0, 0) {$\ket{\Phi^+_\text{A}}$};
		\node[qubit] (b) at (0, -1) {$\ket{\psi}$};
		\draw[wire] (a) -- ([shift={(4, 0)}] a.center) node[qubit] {};
		\draw[wire] (b) -- ([shift={(4, 0)}] b.center) node[qubit] {};

		\draw[wire]
		($([shift={(1,0)}] a)!0.5!([shift={(2,0)}] a)$) --
		($([shift={(1,0)}] b)!0.5!([shift={(2,0)}] b)$)
		;
		\draw[wirejoin]
			($([shift={(1,0)}] b)!0.5!([shift={(2,0)}] b)$)
			circle[radius=0.1] coordinate(dot)
		;

		\qubox{b}{2}{b}{3}{$H$}
		\qubox{a}{1}{a}{2}{$X$}
	\end{tikzpicture}
\end{center}
What should Bob do so that $\ket{\Phi^+_B}$ takes the state $\ket{\psi}$ had initially?

\begin{solution}
	\begin{itemize}
		\item
		If Bob receives \texttt{00}, he does nothing.

		\item
		If Bob receives \texttt{01}, he applies an $X$ gate to his qubit.

		\item
		If Bob receives \texttt{01}, he applies a $Z$ gate to his qubit.

		\item
		If Bob receives \texttt{11}, he applies $ZX$ to his qubit.
	\end{itemize}

	\linehack{}

	The complete circuit is shown below. Double lines indicate classical bits.


	\begin{center}
		\begin{tikzpicture}[scale = 1]
			\node[qubit] (a) at (0, -1) {$\ket{\Phi^+_\text{A}}$};
			\node[qubit] (b) at (0, -2) {$\ket{\Phi^+_\text{B}}$};
			\node[qubit] (c) at (0, 0) {$\ket{\psi}$};
			\draw[wire] (a) -- ([shift={(5, 0)}] a.center) node[qubit] {};
			\draw[wire] (b) -- ([shift={(9, 0)}] b.center) node[qubit] {$\ket{\psi}$};
			\draw[wire] (c) -- ([shift={(5, 0)}] c.center) node[qubit] {};

			\draw[wire, double]
				([shift={(5, 0)}] a.center)
				-- ([shift={(9, 0)}] a.center)
				node[qubit] {}
			;

			\draw[wire, double]
				([shift={(5, 0)}] c.center)
				-- ([shift={(9, 0)}] c.center)
				node[qubit] {}
			;

			\draw[wire]
				($([shift={(1,0)}] a)!0.5!([shift={(2,0)}] a)$) --
				($([shift={(1,0)}] c)!0.5!([shift={(2,0)}] c)$)
			;
			\draw[wirejoin]
				($([shift={(1,0)}] c)!0.5!([shift={(2,0)}] c)$)
				circle[radius=0.1] coordinate(dot)
			;

			\qubox{c}{2}{c}{3}{$H$}
			\qubox{a}{1}{a}{2}{$X$}

			\qubox{a}{3.8}{a}{5.5}{measure}
			\qubox{c}{3.8}{c}{5.5}{measure}


			\draw[wire, double]
				($([shift={(6,0)}] a)!0.5!([shift={(7,0)}] a)$) --
				($([shift={(6,0)}] b)!0.5!([shift={(7,0)}] b)$)
			;
			\draw[wirejoin]
				($([shift={(6,0)}] a)!0.5!([shift={(7,0)}] a)$)
				circle[radius=0.1] coordinate(dot)
			;

			\qubox{b}{6}{b}{7}{$X$}


			\draw[wire, double]
			($([shift={(7,0)}] b)!0.5!([shift={(8,0)}] b)$) --
			($([shift={(7,0)}] c)!0.5!([shift={(8,0)}] c)$)
			;
			\draw[wirejoin]
				($([shift={(7,0)}] c)!0.5!([shift={(8,0)}] c)$)
				circle[radius=0.1] coordinate(dot)
			;
			\qubox{b}{7}{b}{8}{$Z$}

		\end{tikzpicture}
	\end{center}
\end{solution}

\vfill
\pagebreak