Write first draft of core concepts and observables subsections

src/thesis/acronyms.tex

@@ -72,3 +72,8 @@
     short=AWGN,
     long=additive white Gaussian noise
 }
+
+\DeclareAcronym{pdf}{
+    short=PDF,
+    long=probability density function
+}

src/thesis/chapters/2_fundamentals.tex

@@ -592,15 +592,30 @@ decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
 \section{Quantum Mechanics and Quantum Information Science}
 \label{sec:Quantum Mechanics and Quantum Information Science}
 
-% TODO: Should the brief intro to QC be made later on or here?
+Designing codes and decoders for \ac{qec} is generally performed on a
+layer of abstraction far removed from the quantum mechanical
+processes underlying the actual qubits.
+Nevertheless, having a fundamental understanding of the related
+quantum mechanical concepts is useful to understand the unique constraints
+of this field.
+The purpose of this section is to convey these concepts to the reader.
+
 %%%%%%%%%%%%%%%%
 \subsection{Core Concepts and Notation}
 \label{subsec:Notation}
 
-\ldots can be very elegantly expressed using the language of
+% Wave functions
-linear algebra.
+
-\todo{Mention that we model the state of a quantum mechanical system
+In quantum mechanics, the evolution of a state of a particle over tme
-as a vector}
+and space is described by a \emph{wave function} $\psi(x,t)$.
+The connection between this function and the world that we can observe
+is the fact that $\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of
+finding a praticle in that particular state.
+
+% Dirac notation
+
+A lot of the related mathematics can be very elegantly expressed
+using the language of linear algebra.
 The so called Bra-ket or Dirac notation is especially appropriate,
 having been proposed by Paul Dirac in 1939 for the express purpose
 of simplifying quantum mechanical notation \cite{dirac_new_1939}.
@@ -611,9 +626,170 @@ For example, two vectors specified by the labels $a$ and $b$
 respectively are written as $\ket{a}$ and $\ket{b}$.
 Their inner product is $\braket{a\vert b}$.
 
+% Expressing wave functions using linear algebra
+
+We can model a wave function $\psi(x,t)$ as a linear combination of different
+\emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as%
+%
+\begin{align*}
+    \psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t)
+    .%
+\end{align*}
+To express this relation using linear algebra, we represent
+$\psi(x,t)$ and $e_n(x,t)$ as vectors $\ket{\psi}$ and $\ket{e_n}$.
+We write%
+%
+\begin{align*}
+    \ket{\psi} = \sum_{n=1}^{\infty}  c_n \ket{e_n}
+    .%
+\end{align*}
+
+% Operators
+
+Another important notion is that of an \emph{operator}, a component
+that takes a function as an input and returns another function as an output.
+Operators are useful to describe the relations between different
+quantities relating to a particle.
+An example of this is the differential operator $\partial x$.
+
+%%%%%%%%%%%%%%%%
+\subsection{Observables}
+\label{subsec:Observables}
+
+% Observable quantities
+
+An \emph{observable quantity} $Q$ is \ldots .
+Due to the probabilistic nature of quantum mechanics, the result of a
+measurement is not deterministic.
+Thus, it is useful to consider the \emph{expected value} $\braket{Q}$
+of an observable quantity in addition to individual measurement results.
+
+% General expression for expected value of observable quantity
+
+If we know the wave function of a particle, we should be able to
+compute $\braket{Q}$ for any observable quantity we wish.
+It can be shown that for any $Q$, we can compute a
+corresponding operator $\hat{Q}$ such that%
+%
+\begin{align}
+    \label{eq:gen_expr_Q_exp}
+    \braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx
+    .%
+\end{align}%
+%
+While the derivation of this relationship is out of the scope of this
+work, we can at least look at an example to illustrate it.
+Considering the position $Q = x$ of a particle and setting the observable
+operator to $\hat{Q} = x$, we can write%
+%
+\begin{align*}
+    \braket{x} = \int_{-\infty}^{\infty} \psi^*(x,t) \cdot x \cdot \psi(x,t) dx
+    = \int_{-\infty}^{\infty} x \lvert \psi(x,t) \rvert ^2 dx
+    .%
+\end{align*}
+%
+Note that $\lvert \psi(x,t) \rvert $ represents the \ac{pdf} of
+finding a particle in a specific state. We immediately see that the
+formula simplifies to the direct calculation of the expected value.
+
+% Determinate states and eigenvalues
+
+% TODO: Introduce determinate states above
+% TODO: Nicer phrasing
+% TODO: Use different symbol for determinate states (not psi)
+% TODO: Fix equation
+Let us now examine how the observable operator $\hat{Q}$ relates to
+the determinate states that make up the overall superposition state
+of the particle.
+We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as%
+%
+\begin{align}
+    \label{eq:gen_expr_Q_exp_lin}
+    \braket{Q} = \braket{\psi \vert \hat{Q}\psi}
+    .%
+\end{align}
+%
+\autoref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
+relationhip.
+The determinate states are inherently deterministic.
+To relate the two, we look at those states $\ket{\psi}$, where the
+variance of the measurements of $Q$ is zero. These are exactly the
+determinate states.%
+%
+\begin{align*}
+    0 &\overset{!}{=} \braket{(Q - \braket{Q})^2}
+    = \braket{\psi \vert (\hat{Q} - \braket{Q})^2 \psi} \\
+    &= \braket{(Q - \braket{Q})\psi \vert (\hat{Q} - \braket{Q}) \psi} \\
+    &= \lVert (Q - \braket{Q}) \psi \rVert^2 \\[3mm]
+    &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{\psi} = 0 \\
+    \label{eq:observable_eigenrelation}
+    &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi}  = \braket{Q} \ket{\psi}
+    .%
+\end{align*}%
+%
+Because we have assumed the variance to be zero, $\braket{Q}$ is now
+the deterministic measurement value corresponding to the determinate
+state $\ket{\psi}$.
+We can see that the determinate states are the \emph{eigenstates} of
+the observable operator $\hat{Q}$ and that the corresponding
+(deterministic) measurement values are the corresponding \emph{eigenvalues}.
+
+% Recap
+
+To summarize, we can mathematically express any observable quantity
+$Q$ using a corresponding operator $\hat{Q}$.
+This operator allows us to both compute the expected value of the
+observable using \autoref{eq:gen_expr_Q_exp_lin}, and describe the
+individual determinate states and corresponding measurement values
+using \autoref{eq:observable_eigenrelation}.
+
+%%%%%%%%%%%%%%%%
+\subsection{Projective Measurements}
+\label{subsec:Projective Measurements}
+
+%%%%%%%%%%%%%%%%
+\subsection{Qubits and Multi-Qubit States}
+\label{subsec:Qubits and Multi-Qubit States}
+
+\red{
+    \begin{itemize}
+        \item Projective measurements
+            \begin{itemize}
+                \item Using the determinate states as a basis for the
+                    superposition state
+                \item The effect of using the obserrvable operator on
+                    the superposition state
+                \item Modelling the observable operator as a series
+                    of projections
+                \item The projection operator and its eigenvalues
+            \end{itemize}
+        \item Qubits and multi-qubit states
+            \begin{itemize}
+                \item The qubit
+                    \begin{itemize}
+                        \item Similar structure to classical
+                            computing: bits are modified with gates
+                            -> quantum bits are modified with quantum gates
+                    \end{itemize}
+                \item The tensor product
+                \item Information is not stored in the individual bit
+                    states but in the correlations / entanglement between them
+                \item -> The size of the vector space
+                \item The X,Z and Y operators
+                \item (?) Notation of operators on multi-qubit states
+            \end{itemize}
+    \end{itemize}
+}
+
+\red{
+    \begin{itemize}
+        \item Representing wave functions as vectors (psi as label,
+            building a vector space using basis functions)
+    \end{itemize}
+}
+
 \red{\textbf{Tensor product}}
 \red{\ldots
-    \todo{Introduce determinate state or use a different word?}
     Take for example two systems with the determinate states $\ket{0}$
     and $\ket{1}$. In general, the state of each can be written as the
     superposition%
@@ -668,11 +844,6 @@ Introducing a new notation for entangled states, we can write%
 \end{align*}
 %
 
-\subsection{Projective Measurements}
-\label{subsec:Projective Measurements}
-
-% TODO: Write
-
 %%%%%%%%%%%%%%%%
 \subsection{Quantum Gates}
 \label{subsec:Quantum Gates}