commit 1eefe32fe45ecae91f4c4b3e74d1db54f8113b67
Author: augustin64 <me.git@augustin64.fr>
Date:   Fri Sep 13 08:59:44 2024 +0200

    Initial commit

diff --git a/.gitignore b/.gitignore
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+*.aux
+*.log
+*.pdf
diff --git a/DM1.tex b/DM1.tex
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+\documentclass[11pt]{exam}
+\usepackage[utf8]{inputenc}
+\usepackage[french]{babel}
+\usepackage[]{amsthm} %lets us use \begin{proof}
+\usepackage{amsmath,amsfonts}
+\usepackage[]{amssymb} %gives us the character \varnothing
+\usepackage[langfont=caps]{complexity}
+\usepackage[T1]{fontenc}
+\usepackage{setspace}
+\usepackage{tcolorbox}
+\usepackage{physics}
+
+\tcbuselibrary{breakable}
+\tcbset{%any default parameters
+  width=0.7\textwidth,
+  halign=justify,
+  center,
+  breakable,
+  colback=white    
+}
+
+\setstretch{1.5}
+
+\title{QCS - DM1}
+\author{Augustin LUCAS}
+\date\today
+%This information doesn't actually show up on your document unless you use the maketitle command below
+
+\begin{document}
+\maketitle %This command prints the title based on information entered above
+
+%Section and subsection automatically number unless you put the asterisk next to them.
+\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}