92 lines
2.2 KiB
TeX
92 lines
2.2 KiB
TeX
|
\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}
|