233 lines
6.3 KiB
TeX
233 lines
6.3 KiB
TeX
\section{One Bit}
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Before we discuss quantum computation, we first need to construct a few tools. \par
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To keep things simple, we'll use regular (usually called \textit{classical}) bits for now.
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\definition{Binary Digits}
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$\mathbb{B}$ is the set of binary digits. In other words, $\mathbb{B} = \{\texttt{0}, \texttt{1}\}$. \par
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\note[Note]{We've seen $\mathbb{B}$ before---it's the set of integers mod 2.}
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\vspace{2mm}
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\definition{Cartesian Products}
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Let $A$ and $B$ be sets. \par
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The \textit{cartesian product} $A \times B$ is the set of all pairs $(a, b)$ where $a \in A$ and $b \in B$. \par
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As usual, we can write $A \times A \times A$ as $A^3$. \par
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\vspace{2mm}
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In this handout, we'll often see the following sets:
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\begin{itemize}
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\item $\mathbb{R}^2$, a two-dimensional plane
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\item $\mathbb{R}^n$, an n-dimensional space
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\item $\mathbb{B}^2$, the set
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$\{(\texttt{0},\texttt{0}), (\texttt{0},\texttt{1}), (\texttt{1},\texttt{0}), (\texttt{1},\texttt{1})\}$
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\item $\mathbb{B}^n$, the set of all possible states of $n$ bits.
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\end{itemize}
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\problem{}
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What is the size of $\mathbb{B}^n$?
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\vfill
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\pagebreak
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% NOTE: this is time-travelled later in the handout.
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% if you edit this, edit that too.
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\cgeneric{Remark}
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Consider a single classical bit. It takes states in $\{\texttt{0}, \texttt{1}\}$, picking one at a time. \par
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The states \texttt{0} and \texttt{1} are fully independent. They are completely disjoint; they share no parts. \par
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We'll therefore say that \texttt{0} and \texttt{1} \textit{orthogonal} (or equivalently, \textit{perpendicular}). \par
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\vspace{2mm}
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We can draw $\vec{e}_0$ and $\vec{e}_1$ as perpendicular axis on a plane to represent this:
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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\fill[color = black] (0, 0) circle[radius=0.05];
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\draw[->] (0, 0) -- (1.5, 0);
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\node[right] at (1.5, 0) {$\vec{e}_0$ axis};
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\fill[color = oblue] (1, 0) circle[radius=0.05];
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\node[below] at (1, 0) {\texttt{0}};
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\draw[->] (0, 0) -- (0, 1.5);
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\node[above] at (0, 1.5) {$\vec{e}_1$ axis};
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\fill[color = oblue] (0, 1) circle[radius=0.05];
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\node[left] at (0, 1) {\texttt{1}};
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\end{tikzpicture}
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\end{center}
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The point marked $1$ is at $[0, 1]$. It is no parts $\vec{e}_0$, and all parts $\vec{e}_1$. \par
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Of course, we can say something similar about the point marked $0$: \par
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It is at $[1, 0] = (1 \times \vec{e}_0) + (0 \times \vec{e}_1)$, and is thus all $\vec{e}_0$ and no $\vec{e}_1$. \par
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\vspace{2mm}
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Naturally, the coordinates $[0, 1]$ and $[1, 0]$ denote how much of each axis a point \say{contains.} \par
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We could, of course, mark the point \texttt{x} at $[1, 1]$, which is equal parts $\vec{e}_0$ and $\vec{e}_1$: \par
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\note[Note]{
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We could also write $\texttt{x} = \vec{e}_0 + \vec{e}_1$ explicitly. \\
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I've drawn \texttt{x} as a point on the left, and as a sum on the right.
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}
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\null\hfill
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\begin{minipage}{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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\fill[color = black] (0, 0) circle[radius=0.05];
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\draw[->] (0, 0) -- (1.5, 0);
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\node[right] at (1.5, 0) {$\vec{e}_0$ axis};
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\draw[->] (0, 0) -- (0, 1.5);
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\node[above] at (0, 1.5) {$\vec{e}_1$ axis};
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\fill[color = oblue] (1, 0) circle[radius=0.05];
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\node[below] at (1, 0) {\texttt{0}};
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\fill[color = oblue] (0, 1) circle[radius=0.05];
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\node[left] at (0, 1) {\texttt{1}};
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\draw[dashed, color = gray, ->] (0, 0) -- (0.9, 0.9);
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\fill[color = oblue] (1, 1) circle[radius=0.05];
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\node[above right] at (1, 1) {\texttt{x}};
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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\fill[color = black] (0, 0) circle[radius=0.05];
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\fill[color = oblue] (1, 0) circle[radius=0.05];
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\node[below] at (1, 0) {\texttt{0}};
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\fill[color = oblue] (0, 1) circle[radius=0.05];
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\node[left] at (0, 1) {\texttt{1}};
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\draw[dashed, color = gray, ->] (0, 0) -- (0.9, 0.0);
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\draw[dashed, color = gray, ->] (1, 0.1) -- (1, 0.9);
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\fill[color = oblue] (1, 1) circle[radius=0.05];
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\node[above right] at (1, 1) {\texttt{x}};
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill\null
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\vspace{4mm}
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But \texttt{x} isn't a member of $\mathbb{B}$, it's not a valid state. \par
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Our bit is fully $\vec{e}_0$ or fully $\vec{e}_1$. By our current definitions, there's nothing in between. \par
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\note{
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Note that the unit vectors $\vec{e}_0$ and $\vec{e}_1$ form an \textit{orthonormal basis} of the plane $\mathbb{R}^2$.
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}
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\vfill
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\pagebreak
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\definition{Vectored Bits}
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This brings us to what we'll call the \textit{vectored representation} of a bit. \par
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Instead of writing our bits as just \texttt{0} and \texttt{1}, we'll break them into their components: \par
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\null\hfill
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\begin{minipage}{0.48\textwidth}
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\[ \ket{0} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = (1 \times \vec{e}_0) + (0 \times \vec{e}_1) \]
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\end{minipage}
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\hfill
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\begin{minipage}{0.48\textwidth}
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\[ \ket{1} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = (0 \times \vec{e}_0) + (1 \times \vec{e}_1) \]
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\end{minipage}
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\hfill\null
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\vspace{2mm}
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This may seem needlessly complex---and it is, for classical bits. \par
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We'll see why this is useful soon enough.
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\generic{One more thing:}
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The $\ket{~}$ you see in the two expressions above is called a \say{ket,} and denotes a column vector. \par
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$\ket{0}$ is pronounced \say{ket zero,} and $\ket{1}$ is pronounced \say{ket one.} \par
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This is called bra-ket notation. $\bra{0}$ is called a \say{bra,} but we won't worry about that for now.
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\problem{}
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Write \texttt{x} and \texttt{y} in the diagram below in terms of $\ket{0}$ and $\ket{1}$. \par
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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\fill[color = black] (0, 0) circle[radius=0.05];
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\draw[->] (0, 0) -- (1.5, 0);
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\node[right] at (1.5, 0) {$\vec{e}_0$ axis};
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\draw[->] (0, 0) -- (0, 1.5);
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\node[above] at (0, 1.5) {$\vec{e}_1$ axis};
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\fill[color = oblue] (1, 0) circle[radius=0.05];
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\node[below] at (1, 0) {$\ket{0}$};
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\fill[color = oblue] (0, 1) circle[radius=0.05];
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\node[left] at (0, 1) {$\ket{1}$};
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\draw[dashed, color = gray, ->] (0, 0) -- (0.9, 0.9);
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\fill[color = ored] (1, 1) circle[radius=0.05];
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\node[above right] at (1, 1) {\texttt{x}};
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\draw[dashed, color = gray, ->] (0, 0) -- (-0.9, 0.9);
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\fill[color = ored] (-1, 1) circle[radius=0.05];
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\node[above right] at (-1, 1) {\texttt{y}};
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak |