did some work

thesis/Makefile

@@ -23,3 +23,4 @@ clean:
 	-rm main.out
 	-rm main.pdf
 	-rm main.toc
+	-rm main.bbl

thesis/chapters/introduction_qc.tex

@@ -10,39 +10,25 @@ $\ket{\downarrow} \equiv \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$ whic
 is helpful in the following discussion.
 
 A gate operating on a qbit is a unitary operator $G \in U(2)$. One can show that
-$\forall G \in U(2)$ $G$ can be expressed as a product of unitary generator matrices,
+$\forall G \in U(2)$ $G$ can be arbitrarily good as a product of unitary generator matrices\cite[Chapter 4.3]{kaye_ea2007},
 common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with 
 \label{ref:singleqbitgates}
 
 $$X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) $$
 $$Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) $$
 $$H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) $$
-$$R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right) $$
+$$R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)$$
 
 \subsection{$N$ Qbit Systems}
 \label{ref:nqbitsystems}
 
-A system of $N$ identical qbits is described by the kronecker product of the
-single qbit states. A nice definition for this product is:
-
-\begin{definition}
-    
-Let $M$, $N$ be complex matrices, $M = \left(\begin{array}{cc} M_1 & M_2 \\ M_3 & M_4 \end{array}\right)$
-. Then the kronecker product is defined as 
-    
-$$M \otimes N = \left(\begin{array}{cc} M_1 \otimes N & M_2 \otimes N \\ M_3 \otimes N & M_4 \otimes N \end{array}\right)$$
-
-Where for any $c \in \mathbb{C}$ $ c \otimes N := cN$.
-\end{definition}
-
-
 \begin{postulate}
-A $N$ qbit quantum mechanical state is the kronecker product of the $N$ single qbit 
+    A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit 
-state. 
+states. The $N$ qbit operators are the tensor product of the $N$ single qbit operators.
 \end{postulate}
 
 Let $\ket{0}_s := \left(\begin{array}{c} 1 \\ 0 \end{array} \right)$ and $\ket{1}_s := \left(\begin{array}{c} 0 \\ 1 \end{array} \right)$
-be the basis of the one-qbit systems. Then two-qbit states are defined as
+be the basis of the one-qbit systems. Then two-qbit basis states are
 
 $$ \ket{0} := \ket{0b00} := \ket{0}_s \otimes \ket{0}_s :=  \left(\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right)$$
 $$ \ket{1} := \ket{0b01} := \ket{0}_s \otimes \ket{1}_s :=  \left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array} \right)$$
@@ -57,14 +43,15 @@ $$ \ket{\psi} = \sum\limits_{i = 0}^{2^N - 1} c_i \ket{i} $$
 $$ \sum\limits_{i = 0}^{2^N - 1} |c_i|^2 = 1$$
 
 One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
-to generate an arbitrary $N$ qbit gate.
+to generate an arbitrary $N$ qbit gate\cite[Chapter 4.3]{kaye_ea2007}.
 
 
 $$ CX(1, 0) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{array}\right)$$
 $$ CZ(1, 0) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$
 
 
-Where $1$ is the act-qbit and $0$ the control-qbit.
+Where $1$ is the act-qbit and $0$ the control-qbit. In words $CX$ ($CZ$) apply an $X$ ($Z$) gate on the act-qbit,
+if the control-qbit is set.
 
 \subsection{Measurement}
 
@@ -79,3 +66,13 @@ Where $1$ is the act-qbit and $0$ the control-qbit.
 \end{postulate}
 
 Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities.
+
+\begin{corrolary}
+    In general the measurement of a qbit is not invertible, in particular it cannot be represented as a
+    unitary operator.
+\end{corrolary}
+
+\begin{proof}
+    The measuerment in not injective: Measuring both
+    $\ket{0}$ and $\frac{1}{\sqrt{2}}(\ket{0} + \ket{1}$ (can) map to $\ket{0}$.
+\end{proof}

thesis/main.bib

@@ -30,3 +30,17 @@
     author={Simon Anders and Hans J. Briegel},
     year=2005
 }
+@book{
+    wuest1995,
+    title={Höhere Mathematik Für Physiker Teil 1},
+    author={Rainer Wüst},
+    year=1995,
+    publisher={de Gruyter}
+}
+@book{
+    kaye_ea2007,
+    title={An Introduction to Quantum Computing},
+    author={Phillip Kaye, Raymond Laflamme and Michelle Mosca},
+    year=2007,
+    publisher={Oxford University Press}
+}

thesis/main.tex

@@ -14,6 +14,7 @@
 
 \newtheorem{definition}{Definition}
 \newtheorem{postulate}{Postulate}
+\newtheorem{corrolary}{Corrolary}
 
 \title{Development of an Extensible Quantum Computing
 Simulator with a Focus on Simulation in the Graph Formalism }