From b28c482e138f97d24e28ba805ed98d2c40aab2e8 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Sat, 18 Apr 2026 20:33:38 +0200 Subject: [PATCH] Clean up observables section --- src/thesis/chapters/2_fundamentals.tex | 178 +++++++++++++++---------- 1 file changed, 107 insertions(+), 71 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index 935f1cb..0882698 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/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 -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. +In quantum mechanics, the state of a particle is described by a +\emph{wave function} $\psi(x,t)$. +The connection between this function and the observable world +is Born's statistical interpretation: +$\lvert \psi (x,t) \rvert^2$ 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 -\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*} +The connection we will make between quantum mechanics and linear +algebra is that we will model the state space of a system as a +\emph{function space}. +We will represent the state of a particle with wave function +$\psi(x,t)$ using the vector $\ket{\psi}$ +\cite[Sec.~3.3]{griffiths_introduction_1995}. + +% 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. +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 \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 . -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. +An \emph{observable quantity} $Q(x,p,t)$ is a quantity of a quantum +mechanical system that we can measure, such as the position $x$ or +momentum $p$ of a particle. +In general, such measurements are not deterministic, i.e., +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 -of the particle. -We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as% +the determinate states of the observable quantity. +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 -variance of the measurements of $Q$ is zero. These are exactly the -determinate states.% +To relate the two, we note that since determinate states should +always yield the same measurement results, the variance of the +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{(Q - \braket{Q})\psi \vert (\hat{Q} - \braket{Q}) - \psi} \nonumber\\ - &= \lVert (Q - \braket{Q}) \psi \rVert^2 \nonumber\\[3mm] - &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{\psi} = + = \braket{e_n \vert (\hat{Q} - \braket{Q})^2 e_n} \nonumber\\ + &= \braket{(Q - \braket{Q})e_n \vert (\hat{Q} - \braket{Q}) + e_n} \nonumber\\ + &= \lVert (Q - \braket{Q}) e_n \rVert^2 \nonumber\\[3mm] + &\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{e_n} = 0 \nonumber\\ \label{eq:observable_eigenrelation} - &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi} - = \underbrace{\braket{Q}}_{\lambda_n} \ket{\psi} + &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{e_n} + = \underbrace{\braket{Q}}_{\lambda_n} \ket{e_n} .% \end{align}% % -Because we have assumed the variance to be zero, $\braket{Q}$ is now -the deterministic measurement value corresponding to the determinate -state $\ket{\psi}$. +Because we have assumed the variance to be zero, the expected value +$\braket{Q}$ is now the deterministic measurement result +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 -(deterministic) measurement values are the corresponding -\emph{eigenvalues} $\lambda_n$. +the observable operator $\hat{Q}$ and that the measurement values are +the corresponding \emph{eigenvalues} $\lambda_n$ +\cite[Postulate~3]{griffiths_introduction_1995}. % Determinate states as a basis -% TODO: Rephrase -% TODO: Show that |c_n|^2 is the probability of finding a particle in -% a given state -% In particular, using the determinate states $\ket{e_n}$ as a basis to -% write the superimposed state -% \begin{align*} -% \ket{\psi} = \sum_{n=1}^{\infty} c_n \ket{e_n} -% , -% \end{align*} +As we are modelling the wave function $\psi(x,t)$ as a vector +$\ket{\psi}$, we can find a set of basis vectors to decompose it into. +We can use the determinate states for this purpose, expressing the state as% +\footnote{ + We are only considering the case of having a \emph{discrete + spectrum} here, i.e., having a discrete set of eigenvalues and vectors. + For continuous spectra, the procedure is analogous. +} +\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 -% TODO: Mention eigenstates and eigenvalues again -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}. +To summarize, we mathematically model an observable quantity +$Q(x,t,p)$ using a corresponding operator $\hat{Q}$, which allows us +to compute the expected value as $\braket{Q} = \braket{\psi +\vert \hat{Q} \psi}$. +The eigenvectors of $\hat{Q}$ are the determinate states +$\ket{e_n},~n\in \mathbb{N}$ and the eigenvalues are the respective +measurement outcomes. +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}.