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\begin{document}

\title{\vspace{-10mm}Introduction to Quantum Computation, UPB\\Winter 2022, Assignment 4\\{\large To be completed by: Thursday, November 10, start of tutorial}}
\date{}
\maketitle

\section{Exercises}
\begin{enumerate}
    \item
     \begin{enumerate}
        \item  Let $A,B\in\LL(\complex^d)$ be positive semi-definite matrices. Prove that $A+B$ is positive semi-definite.
        \item  Prove that if $\rho$ and $\sigma $ density matrices, then so is $p_1\rho+p_2\sigma$ for any $p_1,p_2\geq 0$ and $p_1+p_2=1$.
     \end{enumerate}
     \item
        Suppose that with probability $1/3$, I give you state $\ket{0}\in\complex^2$, and with probability $2/3$, I give you state $\ket{-}$. Write down (i.e. as a $2\times 2$ matrix) the density matrix describing the state in your possession.

     \item  Define bipartite state $\ket{\psi}=\alpha\ket{01}-\beta\ket{10}$. Let $\rho=\frac{1}{2}\ketbra{\Phi^+}{\Phi^+} + \frac{1}{2}\ketbra{\psi}{\psi}$. Compute $\trace_B(\rho)$. 

     \item  Let $\ket{\psi}=\alpha_0\ket{a_0}\ket{b_0}+\alpha_1\ket{a_1}\ket{b_1}$ be the Schmidt decomposition of a two-qubit state $\ket{\psi}$. Prove that for any single qubit unitaries $U$ and $V$, $\ket{\psi}$ is entangled if and only if $\ket{\psi'}=(U\otimes V)\ket{\psi}$ is entangled. (Hint: Prove that the Schmidt rank of $\ket{\psi}$ equals that of $\ket{\psi'}$. Also, you might find Lemma 1 of the Lecture 3 notes useful.)
\end{enumerate}

\end{document}