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\begin{document}
\openup 1 \jot


\noindent
\bf\noindent
{\large\bf Some macros for typesetting quantum computing homework
\hfill C.K. Li}

\rm





\begin{itemize}

\item In quantum physics, we use the unit vectors
$|0\ra = \begin{pmatrix} 1 \cr 0 \cr\end{pmatrix}$ and 
$|1\ra = \begin{pmatrix} 0 \cr 1 \cr\end{pmatrix}$ to represent 
the states.


\medskip
\centerline{
$|0\ra \la 0| = \begin{pmatrix} 1 \cr 0 \cr \end{pmatrix}(1 \ 0) =
\begin{pmatrix} 1& 0 \cr 0 & 0 \cr\end{pmatrix} = E_{11}$ \quad
and $\quad 
|1\ra \la 1| =  \begin{pmatrix} 0 \cr 1 \cr \end{pmatrix} (0 \ 1) =
\begin{pmatrix} 0& 0 \cr 0 & 1 \cr\end{pmatrix} = E_{22}$. 
}

\item A photon in a quantum environment has the form 
$|\psi \ra = \begin{pmatrix} a \cr b \cr \end{pmatrix}$ 
with $a, b \in \IC$ such that $|a|^2 + |b|^2 = 1$.
Note that we have to use complex numbers!


\item In matrix form, we have $\rho =  |\psi \ra \la \psi| = 
\begin{pmatrix} |a|^2 & a\bar b\cr b\bar a & |b|^2\cr\end{pmatrix}$.
Note that $\rho = \rho^\dag$.

\item Example: If $|\psi\ra = \sqrt{3}
\begin{pmatrix} 1 \cr 1+i \cr \end{pmatrix}$,
then $|\psi \ra \la \psi|$.

\item Suppose $A = E_{11}, B = E_{22}, C = \frac{1}{2} \begin{pmatrix}
1 & 1 \cr 1 & 1 \cr
\end{pmatrix}$.
Then $ABC = 0 = CAB$, but 
$ACB \ne 0$. 

\item Note that
$C = \frac{1}{2} \begin{pmatrix}
1 & 1 \cr 1 & 1 \cr
\end{pmatrix}$ and  
$D = \frac{1}{2} \begin{pmatrix} 1 & -1 \cr -1 & 1 \cr\end{pmatrix}$
are two measurable states of photons 
using a different basis/frames. 


\item $\IC^n$ is a vector space under addition
and multiplication. 

$v_1 + v_2$ for $v_1, v_2 \in \IC^n,$
$\mu v$ for $\mu \in \IC, v \in \IC^n$.



\item 
In particular, $\la v|$ is the conjugate
transpose of $|v\ra$.

\item It is easy to 
express a vector as a linear combination of 
orthonormal basis $\{|e_1\ra, \dots, |e_n\ra\}$.

\item The set $\{P_j = |e_j\ra \la e_j|: j = 1, \dots, n\}$ form a complete set of 
projection operators/matrices.

\item Gram-Schmidt orthogonalization/orthonormalization.


Let $\{|x_1\ra, \dots, |x_m\ra\}$ be linearly independent.

Set $|e_1\ra = |x_1\ra/\| |x_1\ra\|$.

Set $|e_2\ra = |f_2\ra/\| |f_2\ra\|$, where
$|f_2\ra = |x_2\ra - \la e_1| x_2\ra |e_1\ra$ is orthogonal to 
$|e_1\ra$. 

For $k > 1$, set $|f_k\ra/\| |f_k\ra\|$, where
$|f_k\ra = |x_k\ra - (\sum_{j=1}^{k-1} \la e_j| x_k\ra |e_j\ra$ is orthogonal to 
$|e_1\ra, \dots, |e_k\ra$.

\item For any basis $\{|x_1\ra, \dots, |x_n\ra\}$ of $\IC^n$,
there is an orthonormal basis $\{|e_1\ra, \dots, |e_n\ra\}$
such that $\{|x_1\ra, \dots, |x_m\ra\}$ and 
$\{|e_1\ra, \dots, |e_m\ra\}$ span the same subspace for 
$m = 1, \dots, n$.

\item For any linearly independent set 
$\{|x_1\ra, \dots, |x_m\ra\}$, we can
get an orthonormal basis $\{|e_1\ra, \dots, |e_n\ra\}$
such that $\{|x_1\ra, \dots, x_k\ra\}$ and 
$\{|e_1\ra, \dots, |e_k\ra\}$ span the same subspace
for $k = 1, \dots, m$.


\end{itemize}
\end{document}






