From 3407aab2deade4ca4a4573a3be9cd317867b1f87 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Sat, 18 Apr 2026 17:51:13 +0200 Subject: [PATCH] Add first draft of projective measurements --- src/thesis/chapters/2_fundamentals.tex | 111 ++++++++++++++++++------- 1 file changed, 82 insertions(+), 29 deletions(-) diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index e30d2e3..935f1cb 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -90,7 +90,6 @@ We can arrange the coefficients of these equations in a \textit{parity-check matrix} (\acs{pcm}) $\bm{H} \in \mathbb{F}_2^{(n-k) \times n}$ and equivalently define the code as \cite[Sec.~3.1.1]{ryan_channel_2009} -% \begin{align*} \mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n : \bm{H}\bm{x}^\text{T} = \bm{0} \right\} @@ -630,7 +629,6 @@ Their inner product is $\braket{a\vert b}$. 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) .% @@ -638,7 +636,6 @@ We can model a wave function $\psi(x,t)$ as a linear combination of different 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} .% @@ -670,25 +667,21 @@ 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 +Note that $\lvert \psi(x,t) \rvert^2 $ 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. @@ -702,40 +695,55 @@ 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*} +\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 \\ + = \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} = + 0 \nonumber\\ \label{eq:observable_eigenrelation} - &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi} = \braket{Q} \ket{\psi} + &\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi} + = \underbrace{\braket{Q}}_{\lambda_n} \ket{\psi} .% -\end{align*}% +\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}. +(deterministic) measurement values are the corresponding +\emph{eigenvalues} $\lambda_n$. + +% 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*} % 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 @@ -747,22 +755,67 @@ using \autoref{eq:observable_eigenrelation}. \subsection{Projective Measurements} \label{subsec:Projective Measurements} +% Projective measurements + +% TODO: Better introduce the collapse of the superposition 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}. +For these, certain restrictions such as repeatability apply: after +measuring a quantum state and thus collapsing it onto one of the +determinate states, futher measurements should yield the same value. +More general methods of modelling measurements exist, e.g., describing +destructive measurements, but they are not relevant to us here +\cite[Box~2.5]{nielsen_quantum_2010}. + +% Projection operators + +% TODO: Fix notational issues related to e_n +We can model the collapse of the original state onto one of the +superimposed basis states as a \emph{projection}. +To see this, we insert \autoref{eq:determinate_basis} into +\autoref{eq:observable_eigenrelation}, obtaining% +\begin{align*} + \hat{Q}\ket{\psi} = \sum_{n=1}^{\infty} c_n \hat{Q} \ket{e_n} + = \sum_{n=1}^{\infty} \lambda_n c_n \ket{e_n} + .% +\end{align*}% +We see that $\hat{Q}$ has the effect of multiplying the component +along each basis vector with the corresponding eigenvalue. +We decompose $\hat{Q}$ into its constituent parts that act on each of +the separate components as +\begin{align*} + \hat{Q} = \sum_{n=1}^{\infty} \lambda_n \hat{P}_n +\end{align*} +using \emph{projection operators} +\begin{align*} + \hat{P}_n := \ket{e_n}\bra{e_n}, \hspace{3mm} n\in \mathbb{N} + . +\end{align*}% +These project a vector onto the subspace spanned by $\ket{e_n}$. + +% Using projection operators to measure if a state has a component +% along a basis vector + +A particularly interesting property of projection operators is that +\begin{align*} + \hat{P}_n (\hat{P}_n \ket{\psi}) = \hat{P}_n^2 \ket{\psi} + = \hat{P}_n \ket{\psi}, +\end{align*}% +and the only way this can hold for any $\ket{\psi}$ is if $\hat{P}_n$ +only has the eigenvalues $0$ or $1$. +As explained in the previous section, the eigenvalues are the results +of performing a measurement. +We can thus use the projection operator as an observable and treat +the eigenvalue as an indicator of the state having a component along +the related basis vector. + %%%%%%%%%%%%%%%% \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