figures + new section

content.tex

@@ -159,6 +159,7 @@ A direct consequence of \cref{thm:superpositionsClosedUnderProbabilisticTransiti
 \end{aligned}\end{equation}
 
 \section{Introducing: Linear Algebra}
+\label{sec:stocastic_matrix_model}
 The definitions of \cref{sec:probabilistic_model} fully describe a probabilistic computational model. Unfortunately, working with them can be quite cumbersome. This section will introduce an algebraic apparatus based on the definitions from above, with many helpful tools to describe computations and state evolutions. As some terminology and especially the linear properties of \cref{def:probabilisticTransitionFunction} already suggest the mathematical framework of choice will be linear algebra. Let's start by assessing the components of the model described above. We have:
 \begin{itemize}
   \item States (in superposition)
@@ -170,6 +171,7 @@ As it turns out, all three components and their interactions can be expressed in
 \subsection{The State Space}
 The defining property of a superposition is the probability distribution of its basis states. Given an enumeration all basis states the superposition is fully defined by the list of probabilities $\parens{p_0, p_1, \dots, p_{N-1}}$.
 \begin{definition}[State Spaces of Probabilistic Computations]
+  \label{def:state_space_probabilistic_computation}
   Given a state basis $\mathbf{B}$ of a $n$-bit register, the state space of probabilistic computations on this register is defined as:
   \begin{equation*}
     \mathbf{B}^n \coloneqq \parensc*{\mathbf{b} = \sum_{i=0}^{N-1} p_i \mathbf{i} \:\middle|\: p_i \in \R_+ \:,\: \sum_{i=0}^{N-1} p_i = 1}
@@ -225,5 +227,26 @@ The final component that sill needs to be expressed in the framework of linear a
   \end{itemize}
 \end{definition}
 
-
 \section{Making it Quantum}
+\Cref{sec:stocastic_matrix_model} formulates mathematical tools to algebraically describe an abstract model of probabilistic computations defined in \cref{sec:probabilistic_model}. This section takes a reverse approach. The tools developed in \cref{sec:stocastic_matrix_model} are based on stochastic matrices, which is an obvious choice to model probabilistic state transitions. Unfortunately this model has some shortcomings. This section first highlights these inconveniences and then fixes them. By doing so the model will gain in computational power, demonstrated by the implementation of Deutsch's algorithm. Finally, it will be shown that this extended model is indeed physically realizable.
+
+\subsection{Cleaning Up}
+The straight forward and rather simplistic choice of using probability coefficients in \cref{def:nbitRegister,def:state_space_probabilistic_computation} results in quite unwieldy state objects especially in the linear algebra representation. Of course, the probability mass of a complete sample space must always sum up to 1, demanding the normalization of state vectors by the $\norm{.}_1$ norm. The state space $\mathbf{B}^n$ defined in this way is an affine combination of its basis vectors. For an 1-bit system this corresponds to the line segment from $\mathbf{0}$ to $\mathbf{1}$ (see \cref{fig:affine_comb}). As \cref{thm:state_space_unit_sphere_surface_isomorphism} already suggest, randomized computations could be viewed as rotating a ray around the origin. If computations essentially are rotations, then angles between state vectors seem somewhat important. Of course with $\mathbf{a}, \mathbf{b} \in \mathbf{B}^n$ it would be possible to calculate the angle between both sates by rescaling their dot product by their lengths $\mathbf{0}^t \mathbf{1} \parens{\abs{\mathbf{a}} \abs{\mathbf{b}}}^{-1}$. State vectors with unit length would greatly simplify angle calculations. Then, the dot product would suffice. Fortunately, \cref{thm:state_space_unit_sphere_surface_isomorphism} states that $\mathbf{B}^n$ is isomorphic to a subset of the surface of the unit sphere. Therefore, it should also be possible to represent the state space as vectors with unit length. To distinguish between both representation we will write state vectors with coordinates on the unit sphere as $\ket{b}$. This notation is the standard notation of quantum states. By definition the length of $\ket{b} = \sum_{i=1}^N \alpha_i \ket{b_i}$ is 1. The linear coefficients $\alpha_i$ are not probabilities but so-called probability amplitudes and the Pythagorean theorem states that $1 = \sum_{i=1}^N \alpha_i^2$. This means squaring the amplitudes or taking the square root of probabilities maps between affine combinations of basis vectors and points on the unit sphere in the state space. As it turns out negative amplitudes must be allowed, thus this mapping is ambiguous and NOT an isomorphism. 
+
+\begin{figure}
+  \centering
+  \begin{subfigure}[][][c]{0.4\linewidth}
+    \includestandalone[width=\textwidth]{figures/affine_state_space}
+    \caption{todo}
+    \label{fig:affine_comb}
+  \end{subfigure}
+  \hfill
+  \begin{subfigure}[][][c]{0.5\linewidth}
+    \includestandalone[width=\textwidth]{figures/amplitudes_to_affine}
+    \caption{todo}
+    \label{fig:amplitudes_to_affine}
+  \end{subfigure}
+\end{figure}
+
+\contentsketch{amplitudes not probabilities}
+\contentsketch[caption={length preserving transition matrix}]{Transition matrix is not length preserving. Length preserving matrix: orthogonal matrix -> negative coefficients -> interference -> Deutsch's algorithm (new computational power)}

figures/affine_state_space.tex

@@ -0,0 +1,35 @@
+\documentclass[tikz]{standalone}
+
+\usepackage{tkz-euclide}
+\usepackage{physics}
+
+\begin{document}
+\begin{tikzpicture}[scale=4]
+  \tkzInit[xmax=1,ymax=1,ymin=-0.1,xmin=-0.1]
+  \tkzDrawX[label={}] \tkzDrawY[label={}]
+  \tkzDefPoint(0,0){orig}
+  \tkzDefPoint(1,0){b0}
+  \tkzDefPoint(0,1){b1}
+  \tkzDefPoint(0.3,0){p0}
+  \tkzDefPoint(0,0.7){p1}
+  \tkzDefPoint(0.3,0.7){b}
+  \tkzLabelPoint(b0){$\mathbf{0}$}
+  \tkzLabelPoint[left](b1){$\mathbf{1}$}
+
+  \tkzDrawLine[thin,gray,add = 0 and 1](orig,b)
+  \tkzDrawSegment[thick](b0,b1) 
+  \tkzLabelSegment[right](b0,b1){$\mathbf{B}^1$}
+
+  \tkzDrawSegment[-stealth](orig,b)
+  \tkzLabelSegment(orig,b){$\mathbf{b}$}
+
+  \tkzDrawSegment[dashed](p0,b)
+  \tkzDrawSegment[dashed](p1,b) 
+  \tkzLabelSegment[right](p0,b){$p_0$}
+  \tkzLabelSegment(p1,b){$p_1$}
+
+  \tkzDrawArc(orig,b0)(b1)
+
+  \tkzDrawPoints(b0,b1,b)
+\end{tikzpicture}
+\end{document}

figures/amplitudes_to_affine.tex

@@ -0,0 +1,45 @@
+\documentclass[tikz]{standalone}
+
+\usepackage{tkz-euclide}
+\usepackage{physics}
+
+\begin{document}
+\begin{tikzpicture}[scale=4]
+  \tkzInit[xmax=1,ymax=1,ymin=-1,xmin=-1]
+  \tkzDrawX[right space=0.2, left space=0.2,label={}]
+  \tkzDrawY[up space=0.2, down space=0.2, label={}]
+  \tkzDefPoint(0,0){orig}
+  \tkzDefPoint(1,0){b0}
+  \tkzDefPoint(0,1){b1}
+  \tkzDefPoint(0.3,0){p0}
+  \tkzDefPoint(0,0.7){p1}
+  \tkzDefPoint(0.3,0.7){b}
+  \tkzDefPoint(0.5477,-0.8367){kb}
+  \tkzDefPoint(0.5477,0){a0}
+  \tkzDefPoint(0,-0.8367){a1}
+  \tkzDrawCircle[thick,gray](orig,b0)
+  \tkzLabelPoint(b0){$\ket{0}$}
+  \tkzLabelPoint[left](b1){$\ket{1}$}
+
+  \tkzDrawSegment[thick,gray](b0,b1) 
+  \tkzLabelSegment[right,gray](b0,b1){$\mathbf{B}^1$}
+
+  \tkzDrawSegment[-stealth](orig,kb)
+  \tkzDrawSegment[dashed](a0,kb)
+  \tkzLabelPoint(a0){$\alpha_0$}
+  \tkzDrawSegment[dashed](a1,kb)
+  \tkzLabelPoint(a1){$\alpha_1$}
+  \tkzLabelSegment(orig,kb){$\ket{b}$}
+
+  \tkzDrawSegment[-stealth](orig,b)
+  \tkzLabelSegment(orig,b){$\mathbf{b}$}
+
+  \tkzDrawSegment[dashed](p0,b)
+  \tkzDrawSegment[dashed](p1,b) 
+  \tkzLabelSegment[right](p0,b){$\alpha_0^2$}
+  \tkzLabelSegment[below](p1,b){$\alpha_1^2$}
+
+
+  \tkzDrawPoints(b0,b1,b,kb)
+\end{tikzpicture}
+\end{document}

main.tex

@@ -8,9 +8,18 @@
 \usepackage[mathscr]{euscript}
 \usepackage{mathtools}
 \usepackage{physics}
+\usepackage[colorinlistoftodos]{todonotes}
 
 \usepackage{cleveref}
 
+\usepackage{tikz}
+\usepackage{standalone}
+\usepackage{tkz-euclide}
+
+\usepackage{caption}
+\usepackage{subcaption}
+
+\newcommand{\contentsketch}[2][]{\todo[inline, color=yellow, #1]{#2}}
 
 \newcommand{\ptrans}{\delta}
 \DeclarePairedDelimiter{\parens}{\lparen}{\rparen}