From 07448c37b937fe2d4f9cd70455ad2f61f51d6e56 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <andreas.tsouchlos@student.kit.edu>
Date: Wed, 19 Nov 2025 10:43:49 +0100
Subject: [PATCH] Change paragraphs; add TODO; change margins

---
 src/intro/main.tex | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/src/intro/main.tex b/src/intro/main.tex
index 70c3971..3833502 100644
--- a/src/intro/main.tex
+++ b/src/intro/main.tex
@@ -1,6 +1,6 @@
 \documentclass[dvipsnames]{article}
 
-\usepackage[a4paper,left=3cm,right=2cm,top=2.5cm,bottom=2.5cm]{geometry}
+\usepackage[a4paper,left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
 \usepackage{float}
 \usepackage{amsmath}
 \usepackage{amsfonts}
@@ -17,6 +17,7 @@
     style=ieee,
     sorting=nty,
 ]{biblatex}
+\usepackage{todonotes}
 
 \usetikzlibrary{calc, positioning}
 
@@ -89,7 +90,6 @@ $E$ is an operator describing a possible error and $E
 \ket{\psi}_\text{L}$ is the resulting state after that error.
 By measuring the corresponding eigenvalue, we can determine if
 $E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
-
 To do this without directly measuring (and thus disturbing) the
 logical state $\ket{\psi}_\text{L}$, we prepare an ancilla
 qubit with state $\ket{0}_\text{A}$ and we entangle it with
@@ -117,18 +117,21 @@ the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in
 \mathcal{F}$, we will deterministically measure $\ket{1}_\text{A}$.
 In general, however, the resulting state of the three-qubit system will be a
 superposition of the two cases.
+
 Note that the expressions $P_\mathcal{C}$ and $P_\mathcal{F}$ above
 essentially constitute projection operators onto $\mathcal{C}$ and
 $\mathcal{F}$. E.g., $P_\mathcal{C}$ will eliminate all
 components of a superposition state $E \ket{\psi}_\text{L}$ that lie
 in $\mathcal{F}$.
-
 By measuring the ancilla qubit, we collapse the overall state
 into one of two configurations. In each of those configurations, $E
 \ket{\psi}_\text{L}$ is projected back onto either $\mathcal{C}$ or
 $\mathcal{F}$. At the same time, because $Z_1Z_2 \ket{\psi}_\text{L}
 = \ket{\psi}_\text{L}$, the projections leave the logical state
-$\ket{\psi}_\text{L}$ untouched. We have thus managed to determine
+$\ket{\psi}_\text{L}$ untouched.
+\todo{Explain that the collapse of the error superposition enables
+the digitization of arbitrary error operations \cite{nielsen_quantum_2010}}
+We have thus managed to determine
 whether an error occurred without disturbing the encoded
 quantum information.