first quantum definitions

content.tex

@@ -375,7 +375,7 @@ O_f = \begin{pmatrix}
 $
 \end{tabular}
 \end{center}
-So it is indeed possible to evaluate a function $f : \parensc{0, 1} \to \parensc{0, 1}$ in the orthogonal computational model. If the second qubit $\ket{y}$ is chosen initialized as $\ket{0}$, measuring that qubit in the computational basis after the application of $O_f$ returns the value of $f(x)$. The trick that makes it possible to anwser $f(0) \stackrel{?}{=} f(1)$ with one call to $O_f$ is to initialize both qubits in a perfect superposition of $\ket{0}$ and $\ket{y}$. Then, both values of $f(0)$ and $f(1)$ will interfere $\ket{y}$.
+So it is indeed possible to evaluate a function $f : \parensc{0, 1} \to \parensc{0, 1}$ in the orthogonal computational model. If the second qubit $\ket{y}$ is initialized as $\ket{0}$, measuring that qubit in the computational basis after the application of $O_f$ returns the value of $f(x)$. The trick that makes it possible to answer $f(0) \stackrel{?}{=} f(1)$ with one call to $O_f$ is to initialize both qubits in a perfect superposition of $\ket{0}$ and $\ket{1}$. Then, both values of $f(0)$ and $f(1)$ will interfere $\ket{y}$.
 \begin{center}
 \begin{quantikz}
   \lstick{$\ket{0}$} & \gate{H}\slice{$\ket{z_1}$} & \gate[2]{O_f}\slice{$\ket{z_2}$} & \gate{H} & \meter{}\\
@@ -415,7 +415,42 @@ That is a lot to digest, but what happened is: Both qubits interfered with each
    &\frac{1}{2}\parens*{\parens*{(-1)^{f(0)} + (-1)^{f(1)}}\ket{0} + \parens*{(-1)^{f(0)} - (-1)^{f(1)}}\ket{1}} = \\
    &\frac{1}{2}(-1)^{f(0)}\parens*{\parens*{1 + (-1)^{f(0) \oplus f(1)}}\ket{0} + \parens*{1 - (-1)^{f(0) \oplus f(1)}}\ket{1}}
 \end{align*}
-Now measuring the first bit returns $\ket{0}$ iff $f(0) = f(1)$ and otherwise $\ket{1}$.
+Now measuring the first bit returns $\ket{0}$ iff $f(0) = f(1)$ and otherwise $\ket{1}$. 
+
+Apparently interference of amplitudes is truly a remarkable feature. There is a key difference between probabilistic and orthogonal computations. In the probabilistic model the concept of computation superposition merely models the lack of knowledge about the true state of the system. If a system is in a probabilistic superposition of two states the true state is just unknown but the system is only in one of the two states at any time. The interference effects in orthogonal computations paint a different picture. In order to interfere with each other the two sates in a orthogonal superposition must somehow be present at the same time. The system has to be in both states, resulting a kind of inherent parallelism built into the model! In Deutsch's algorithm, the application of $O_f$ to a superposition does in fact not equal one call to $f$ but evaluating $f$ in parallel for both inputs. 
 \todo[inline]{rename qubits as the term is not yet defined}
 
+\subsection{Entering the Real World: Quantum Computing}
+The best theoretical model is of no value if it relies on some kind of magic that not be realized in the real world. Luckily if the real probability amplitudes of the orthogonal model are replaced by complex numbers, one ends up with a computational model which can perfectly described by quantum mechanics, thus making it a real world physical system, fittingly called \emph{quantum computing}. Of course, real probability amplitudes are a subset of complex probably amplitudes and likewise orthogonal computations are a subset of quantum computations. Only minor adjustments need to be made to the mathematical framework to handle complex amplitudes. The fundamental unit of information is a \emph{qubit} this special terminology is intended to highlight the special properties of quantum superpositions. Also, oftentimes greek leters are used in the quantum case.
 
+\begin{definition}[Qubit]
+  \label{def:qubit}
+  A qubit is a quantum superposition with unit length of two orthonormal basis vectors $\ket{0}$ and $\ket{1}$:
+  \begin{equation}
+    \ket{\psi} = \alpha \ket{0} + \beta \ket{1}
+  \end{equation}
+  With the transposed complex conjugate of $\ket{\psi}$ denoted as:
+  \begin{equation}
+    \parens*{\ket{\psi}^*}^t = \ket{\psi}^\dagger = \bra{\psi} = \bra{0} \alpha^* + \bra{1} \beta^*
+  \end{equation}
+  Furthermore, every qubit satisfies the normalization criterion:
+  \begin{equation}
+    \braket{\psi} = \alpha^*\alpha + \beta^*\beta = \abs{\alpha}^2 + \abs{\beta}^2 = 1
+  \end{equation}
+\end{definition}
+\begin{remark}
+  In accordance with the standard in quantum computing literature the computational basis states $\ket{0}$ and $\ket{1}$ where chosen for in \cref{def:qubit}. However, similarly to orthogonal matrices, unitary matrices (their complex extension) also map any complex orthonormal basis to another complex orthonormal basis of the same dimension. Also, every bijective mapping of two complex orthonormal bases is a unitary operation. So, any qubit can be transformed to another basis by simply applying a unitary transformation to it, making the basis choice of a qubit irrelevant.
+\end{remark}
+  
+\begin{definition}[Computational Basis]
+  The standard bais of quantum computing is the computational basis. The basis $2^n$ values of a $n$-bit register are mapped to an orthonormal basis $\mathcal{B} = \parensc*{\ket{i}}_{i=0}^{2^n - 1}$, with $i$ being the decimal value of the $n$-bit register. In coordinate space $\ket{i}$ is represented by $e_i \in \C^{2^n}$:
+  \begin{equation}
+    \ket{i} \cong e_i = \parens{\underbrace{0,\dots,0}_{i - 1},1,0,\dots,0}^t
+  \end{equation}
+\end{definition}
+
+\begin{definition}[Quantum State Space]
+\begin{equation*}
+  \mathcal{B}^n = \parensc*{\ket{\psi} = \sum_{i=0}^{2^n -1} \alpha_i \ket{i} \:\middle|\: \alpha_i \in \C ,\: \braket{\psi} = 1}
+\end{equation*}
+\end{definition}

main.tex

@@ -31,6 +31,7 @@
 
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