Clean up observables section

src/thesis/chapters/2_fundamentals.tex

@@ -593,9 +593,9 @@ decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
 
 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.
+processes underlying the actual physics.
 Nevertheless, having a fundamental understanding of the related
-quantum mechanical concepts is useful to understand the unique constraints
+quantum mechanical concepts is useful to grasp the unique constraints
 of this field.
 The purpose of this section is to convey these concepts to the reader.
 
@@ -605,11 +605,12 @@ The purpose of this section is to convey these concepts to the reader.
 
 % Wave functions
 
-In quantum mechanics, the evolution of a state of a particle over tme
+In quantum mechanics, the state of a particle is described by a
-and space is described by a \emph{wave function} $\psi(x,t)$.
+\emph{wave function} $\psi(x,t)$.
-The connection between this function and the world that we can observe
+The connection between this function and the observable world
-is the fact that $\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of
+is Born's statistical interpretation:
-finding a praticle in that particular state.
+$\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of finding a praticle at
+position $x$ and time $t$ \cite[Sec.~1.2]{griffiths_introduction_1995}.
 
 % Dirac notation
 
@@ -627,24 +628,32 @@ 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
+The connection we will make between quantum mechanics and linear
-\emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as%
+algebra is that we will model the state space of a system as a
-\begin{align*}
+\emph{function space}.
-    \psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t)
+We will represent the state of a particle with wave function
-    .%
+$\psi(x,t)$ using the vector $\ket{\psi}$
-\end{align*}
+\cite[Sec.~3.3]{griffiths_introduction_1995}.
-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 can model a wave function $\psi(x,t)$ as a linear combination of different
-We write%
+% \emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as%
-\begin{align*}
+% \begin{align*}
-    \ket{\psi} = \sum_{n=1}^{\infty}  c_n \ket{e_n}
+%     \psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t)
-    .%
+%     .%
-\end{align*}
+% \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
+Another important notion is that of an \emph{operator}, a transformation
-that takes a function as an input and returns another function as an output.
+that takes a function as an input and returns another function as an
+output \cite[Sec.~3.2.2]{griffiths_introduction_1995}.
 Operators are useful to describe the relations between different
 quantities relating to a particle.
 An example of this is the differential operator $\partial x$.
@@ -655,18 +664,22 @@ An example of this is the differential operator $\partial x$.
 
 % Observable quantities
 
-An \emph{observable quantity} $Q$ is \ldots .
+An \emph{observable quantity} $Q(x,p,t)$ is a quantity of a quantum
-Due to the probabilistic nature of quantum mechanics, the result of a
+mechanical system that we can measure, such as the position $x$ or
-measurement is not deterministic.
+momentum $p$ of a particle.
-Thus, it is useful to consider the \emph{expected value} $\braket{Q}$
+In general, such measurements are not deterministic, i.e.,
-of an observable quantity in addition to individual measurement results.
+measurements on identically prepared states can yield different results.
+There are some states, however, that are \emph{determinate} for a
+specific observable: measuring those will always yield identical
+observations \cite[Sec.~3.3]{griffiths_introduction_1995}.
 
 % 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.
+compute the expected value $\braket{Q}$ of any observable quantity we wish.
 It can be shown that for any $Q$, we can compute a
-corresponding operator $\hat{Q}$ such that%
+corresponding operator $\hat{Q}$ such that
+\cite[Sec.~3.3]{griffiths_introduction_1995}
 \begin{align}
     \label{eq:gen_expr_Q_exp}
     \braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx
@@ -675,7 +688,8 @@ corresponding operator $\hat{Q}$ such that%
 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%
+operator to $\hat{Q} = x$, we can write
+\cite[Sec.~1.5]{griffiths_introduction_1995}
 \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
@@ -687,14 +701,10 @@ 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
+the determinate states of the observable quantity.
-of the particle.
+We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as
-We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as%
+\cite[Eq.~3.114]{griffiths_introduction_1995}
 \begin{align}
     \label{eq:gen_expr_Q_exp_lin}
     \braket{Q} = \braket{\psi \vert \hat{Q}\psi}
@@ -703,53 +713,79 @@ We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as%
 \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
+To relate the two, we note that since determinate states should
-variance of the measurements of $Q$ is zero. These are exactly the
+always yield the same measurement results, the variance of the
-determinate states.%
+observable should be zero.
+We thus compute \cite[Eq.~3.116]{griffiths_introduction_1995}
 \begin{align}
     0 &\overset{!}{=} \braket{(Q - \braket{Q})^2}
-    = \braket{\psi \vert (\hat{Q} - \braket{Q})^2 \psi} \nonumber\\
+    = \braket{e_n \vert (\hat{Q} - \braket{Q})^2 e_n} \nonumber\\
-    &= \braket{(Q - \braket{Q})\psi \vert (\hat{Q} - \braket{Q})
+    &= \braket{(Q - \braket{Q})e_n \vert (\hat{Q} - \braket{Q})
-    \psi} \nonumber\\
+    e_n} \nonumber\\
-    &= \lVert (Q - \braket{Q}) \psi \rVert^2 \nonumber\\[3mm]
+    &= \lVert (Q - \braket{Q}) e_n \rVert^2 \nonumber\\[3mm]
-    &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{\psi} =
+    &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{e_n} =
     0 \nonumber\\
     \label{eq:observable_eigenrelation}
-    &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi}
+    &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{e_n}
-    = \underbrace{\braket{Q}}_{\lambda_n} \ket{\psi}
+    = \underbrace{\braket{Q}}_{\lambda_n} \ket{e_n}
     .%
 \end{align}%
 %
-Because we have assumed the variance to be zero, $\braket{Q}$ is now
+Because we have assumed the variance to be zero, the expected value
-the deterministic measurement value corresponding to the determinate
+$\braket{Q}$ is now the deterministic measurement result
-state $\ket{\psi}$.
+corresponding to the determinate state
+$\ket{e_n},~n\in \mathbb{N}$.
 We can see that the determinate states are the \emph{eigenstates} of
-the observable operator $\hat{Q}$ and that the corresponding
+the observable operator $\hat{Q}$ and that the measurement values are
-(deterministic) measurement values are the corresponding
+the corresponding \emph{eigenvalues} $\lambda_n$
-\emph{eigenvalues} $\lambda_n$.
+\cite[Postulate~3]{griffiths_introduction_1995}.
 
 % Determinate states as a basis
 
-% TODO: Rephrase
+As we are modelling the wave function $\psi(x,t)$ as a vector
-% TODO: Show that |c_n|^2 is the probability of finding a particle in
+$\ket{\psi}$, we can find a set of basis vectors to decompose it into.
-% a given state
+We can use the determinate states for this purpose, expressing the state as%
-% In particular, using the determinate states $\ket{e_n}$ as a basis to
+\footnote{
-% write the superimposed state
+    We are only considering the case of having a \emph{discrete
-% \begin{align*}
+    spectrum} here, i.e., having a discrete set of eigenvalues and vectors.
-%     \ket{\psi} = \sum_{n=1}^{\infty} c_n \ket{e_n}
+    For continuous spectra, the procedure is analogous.
-%     ,
+}
-% \end{align*}
+\begin{align}
+    \label{eq:determinate_basis}
+    \ket{\psi} = \sum_{n=1}^{\infty} c_n \ket{e_n}, \hspace{3mm}
+    c_n := \braket{e_n \vert \psi}
+    .%
+\end{align}
+Inserting \autoref{eq:determinate_basis} into
+\autoref{eq:gen_expr_Q_exp_lin} we obtain
+% tex-fmt: off
+\cite[Prob.~3.35c)]{griffiths_introduction_1995}
+% tex-fmt: on
+\begin{align*}
+    \braket{Q} = \left( \sum_{n=1}^{\infty} c_n \bra{e_n} \right)
+    \left( \sum_{m=1}^{\infty} c_m\hat{Q}\ket{e_m} \right)
+    = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} c_n c_m
+    \lambda_m\braket{e_n \vert e_m}
+    = \sum_{n=1}^{\infty} \lambda_n \lvert c_n \rvert ^2
+    .%
+\end{align*}
+We can thus interpret $\lvert c_n \rvert ^2$ as the probability of
+obtaining value $\lambda_n$ from the measurement.
 
 % Recap
 
-% TODO: Mention that `observable` is used to refer to the observable operator
+To summarize, we mathematically model an observable quantity
-% TODO: Mention eigenstates and eigenvalues again
+$Q(x,t,p)$ using a corresponding operator $\hat{Q}$, which allows us
-To summarize, we can mathematically express any observable quantity
+to compute the expected value as $\braket{Q} = \braket{\psi
-$Q$ using a corresponding operator $\hat{Q}$.
+\vert \hat{Q} \psi}$.
-This operator allows us to both compute the expected value of the
+The eigenvectors of $\hat{Q}$ are the determinate states
-observable using \autoref{eq:gen_expr_Q_exp_lin}, and describe the
+$\ket{e_n},~n\in \mathbb{N}$ and the eigenvalues are the respective
-individual determinate states and corresponding measurement values
+measurement outcomes.
-using \autoref{eq:observable_eigenrelation}.
+We can decompose an arbitrary state as $\ket{\psi} = \sum_{n=1}^{\infty} c_n
+\ket{e_n}$, where $\lvert c_n \rvert ^2$ represents the probability
+of obtaining a certain measurement value.
+Note that when we speak of an \emph{observable}, we are usually
+refering to the corresponding operator $\hat{Q}$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Projective Measurements}
@@ -757,7 +793,7 @@ using \autoref{eq:observable_eigenrelation}.
 
 % Projective measurements
 
-% TODO: Better introduce the collapse of the superposition state
+% TODO: Introduce concept of collapsing the wave function onto a basis state
 The measurements we considered in the previous section, for which
 \autoref{eq:gen_expr_Q_exp_lin} holds, belong to the category of
 \emph{projective measurements}.