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\begin{document}

\title{\vspace{-10mm}Introduction to Quantum Computation, UPB\\Summer 2021, Assignment 3\\{\large To be completed by: Friday, May 7}}
\date{}
\maketitle

\section{Exercises}
\begin{enumerate}
    \item
    \begin{enumerate}
        \item Let $A\in\LL(\complex^d)$ be Hermitian. Prove that if for all $\ket{\psi}\in\complex^d$, $\bra{\psi}A\ket{\psi}\geq 0$, then $A$ has only non-negative eigenvalues. (Hint: Start by taking the spectral decomposition of $A$, and then make clever choices for $\ket{\psi}$.)
        \item Let $A\in\LL(\complex^d)$ be Hermitian. Prove that if $A$ has only non-negative eigenvalues, then for all $\ket{\psi}\in\complex^d$, $\bra{\psi}A\ket{\psi}\geq 0$. (Hint: Write $\ket{\psi}$ with respect to the eigenbasis of $A$.)
    \end{enumerate}
\item

           Let $\ket{\psi}=\ket{-}\in\complex^2$. Suppose we measure in the $Z$ basis $B=\set{\ketbra{0}{0},\ketbra{1}{1}}$. What are the probabilities for each possible measurement outcome, and the corresponding post-measurement states?

  \item Consider the teleportation protocol we saw in class. Does it still work if we replace the use of the entangled Bell state $\ket{\phi^+}$ with the unentangled state $\ket{00}$ (i.e. Alice and Bob share the state $\ket{00}$)? How about if we use $\sqrt{2/5}\ket{00}+\sqrt{3/5}\ket{11}$ instead of $\ket{\phi^+}$?
\end{enumerate}

\end{document}
