\documentclass[11pt]{exam}
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\usepackage{tcolorbox}
\usepackage{physics}

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\title{QCS - DM1}
\author{Augustin LUCAS}
\date\today
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\begin{document}
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\section*{Q1}

\begin{tcolorbox}
A system $X$ is in:

\[v_0 = 
  \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix} \;
  \text{ or }
  v_1 =
  \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix} \;
\]
We want to know which is the case. Describe a sequence of operations that determines which is the case.
\end{tcolorbox}

Let $A = \begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{pmatrix}$

$A$ is a unitary matrix, with $Av_0 = \ket{0}$ and  $Av_1 = \ket{1}$

Apply the following sequence of operations:
\begin{itemize}
  \item Apply transformation $A$
  \item Perform measurement
\end{itemize}

This allows us to distinguish in which of both states was the system initially.

\section*{Q2}

\begin{tcolorbox}
Show that $w = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ 0 \\ \frac{1}{\sqrt{2}} \end{pmatrix} \;$ is not of the form $v \otimes v'$
\end{tcolorbox}

Suppose that $w$ is of form $v \otimes v'$. Given such $v=\begin{pmatrix} a \\ b \end{pmatrix}, v'=\begin{pmatrix} c \\ d \end{pmatrix}$,
we have:

\[
v \otimes v' =
\begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}
\]

Then:
\left \{
\begin{array}{c @{=} c}
    ac & \frac{1}{\sqrt{2}} \\
    ad & 0 \\
    bc & 0 \\
    bd & \frac{1}{\sqrt{2}}
\end{array}
\right.


As $a=0$ or $d=0$, $ac=0$ or $bd=0$. Impossible !

As such, $w$ is not of form $v \otimes v'$

\end{document}