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\begin{document}
\openup 1\jot

\noindent
{\bf Math 410 Quantum Computing \qquad Examination 1}  {\hfill Due: noon, 12 October, 2012}


\bigskip\noindent
{\bf 1.} 
(a) Show that if  $A = (A_{ij}) \in M_n$ is normal, 
then $\sum_{j=1}^n |A_{ij}|^2 = \sum_{j=1}^n |A_{ji}|^2$ for 
$i = 1, \dots, n$.

(b) Construct an example of a non-normal matrix $A = (A_{ij}) \in M_2$ 
 such that  
 
 \medskip
 \hskip 1in $\sum_{j=1}^n |A_{ij}|^2 = \sum_{j=1}^n |A_{ji}|^2$ for 
$i = 1, 2$.

\bigskip\noindent
{\bf 2.} Let 
$$\rho = {1\over 8}\begin{pmatrix}
3 & 0 & 2i & 0 \cr
 0 & 1 & 0  & i \cr
 -2i & 0 & 3 & 0 \cr
0 & -i & 0 & 1 \cr\end{pmatrix} \in M_2 \otimes M_2.$$


(a) Find the eigenvalues of $\rho$
and show that $\rho$ is a mixed state.

 
(b) Compute $\tr_1(\rho)$ and $\tr_2(\rho)$.



\medskip\noindent
{\bf 3.} (a) Suppose $|\psi\ra \in \IC^2$ is a unit vector and 
$\rho = |\psi \ra \la \psi|$.  Show that $\rho|\psi\ra = |\psi\ra$.

(b) Let $|\psi_1\ra = \frac{1}{\sqrt 2}\begin{pmatrix}1\cr i\cr\end{pmatrix}$, 
$|\psi_2\ra = \begin{pmatrix} 1\cr 0\cr\end{pmatrix}$, 
$|\psi_3\ra = \begin{pmatrix} 0\cr 1\cr\end{pmatrix}$, 
and $\rho = \frac{1}{3} \sum_{j=1}^3 |\psi_j\ra \la \psi_j|$.


\hskip .3in Find the spectral 
decomposition of $\rho$ and determine $\log(\rho)$ and $\exp(\rho)$. 



\medskip\noindent
{\bf 4.} Suppose 
$$
\rho_1 =  {1\over 2}\begin{pmatrix}
0 & 0 & 0 & 0 \cr
0 & 1 & 0 & 0 \cr
0 & 0 & 1 & 0 \cr
0 & 0 & 0 & 0 \cr\end{pmatrix} 
\quad\hbox{ and } \quad
\rho_2 = {1\over 2} \begin{pmatrix}
0 & 0 & 0 & 0 \cr
0 & 1 & 1 & 0 \cr
0 & 1 & 1 & 0 \cr
0 & 0 & 0 & 0 \cr\end{pmatrix}.$$
\quad \
(a) Compute the fidelity $F(\rho_1, \rho_2)$.

 
(b) Consider the measurement operator
$$M = \frac{1}{10}
\begin{pmatrix}
 1 & 0 & 0 & i \cr
 0 & 3 & 2+i  & 0 \cr
 0 & 2-i & 3 & 0\cr
 -i & 0 & 0 & 1 \cr\end{pmatrix}.$$
\hskip .3in Determine the quantum state after applying the measurement $M$
to $\rho_2$ in (a).

\medskip\noindent
{\bf 5.} Let $\rho = {1\over 4} \begin{pmatrix}3 & i \cr -i & 1\cr\end{pmatrix}$.

\quad
(a) Show that $\rho$ is a mixed state.

\quad
(b) Determine two different $|\Psi_1\ra, |\Psi_2\ra \in 
\IC^4 \equiv \IC^2\otimes \IC^2$, which are not multiples of each others, 

\qquad such that 
$$\tr_2(\rho_1) = \tr_2(\rho_2) = \rho \quad \hbox{ for }
\rho_j = |\psi_j\ra \la \psi_j| \quad \hbox{ with } j = 1, 2.$$

\newpage
\medskip\noindent
{\bf 6.} Let 
$$
\rho_1 = {1\over 2} \begin{pmatrix} 
\cos^2 t & 0 & 0 & \cos t \sin t \cr
0 & \sin^2 t & \cos t \sin t & 0 \cr
0 & \cos t \sin t & \cos^2 t & 0 \cr
\cos t \sin t & 0 & 0 & \sin^2 t \cr\end{pmatrix}
$$
\qquad and
$$
\rho_2 = {1\over 2} \begin{pmatrix} 
\cos^2 t & 0 & 0 & \cos t \sin t \cr
0 & \sin^2 t & \cos t \sin t & 0 \cr
0 & \cos t \sin t & \sin^2 t & 0 \cr
\cos t \sin t & 0 & 0 & \cos^2 t \cr\end{pmatrix}.$$

(a) Show that $\rho_1$ is separable for all $t \in \IR$.
 
(b) Determine $t \in [0, 2\pi]$ such that $\rho_2$ is separable.



\medskip\noindent
{\bf 7.}  Suppose an $mn$-by-$mn$ density matrix
$A \in M_m \otimes M_n$ is 
in block form $A = (A_{ij})$ with $A_{ij} \in H_n$.
 
Recall that 
$$\tr_1(A) = \sum_{j=1}^m (\la e_j| \otimes I_n) A (|e_j\ra \otimes 
I_n)$$
\qquad
for a fixed orthonormal basis $\{|e_1\ra, \dots, |e_m\ra\}$  for $\IC^m$. 

(a)  
Suppose 
$\{|e_1\ra, \dots, |e_m\ra\}$ 
is the standard basis for $\IC^m$. 

\hskip .3in Show that 
$\la e_1| \otimes I_n = [I_n | 0_n | \dots | 0_n]$,
$\la e_2| \otimes I_n = [0_n | I_n | 0_n | \dots | 0_n]$, etc.


(b) Show that $\tr_1(A) = A_{11} + \cdots + A_{mm}$.

(c) Show that  $\tr_1(A)$ is the same for any choice of 
orthonormal basis $\{|e_1\ra, \dots, |e_m\ra \}$ for $\IC^m$.

(d) Let $\tilde A$ be obtained from $A$ by changing $A_{ij}$ to $0$
whenever $i\ne j$. 
Show that $\tr_1(A) = \tr_1(\tilde A)$.




\medskip\noindent
{\bf 8.} (a) Explain why there is or there is no  unitary matrix $U$
(quantum gate) such that
$$(U|00\ra, U|01\ra, U|10\ra, U|11\ra ) = \left(|00\ra, |01\ra, |10\ra, 
\frac{1}{2}(|00\ra+|01\ra+|10\ra+|11\ra) \right).$$

\quad
(b)  Explain why there is or there is no unitary matrix $V$ 
(quantum gate) such that
$$(V|000\ra, V|111\ra) = (|GHZ\ra, |W\ra) 
= \left(\frac{1}{\sqrt{2}}(|000\ra + |111\ra), 
\frac{1}{\sqrt 3}(|100\ra +|010\ra + |001\ra)\right).$$

\end{document}


 
