Physics 511, Homework 4
Prob. 1. Albers. P59, Prob. 2.8
Prob. 2. Quantum no-cloning theorem: If
states with
1 and 2 represent two arbitrary non-orthogonal
1 2  0 , prove that there is no unitary quantum operation that can copy these two
states, i.e. the evolution
1 0  1 1
2 0  2 2
is impossible, where
0 is an arbitrary state. This theorem makes one of the foundations for
security for quantum cryptography.
Prob.
3.
A
single-qubit
density
matrix
can
always
be
expanded
into
1
1
   n0 I  n1 x  n2 y  n3 z  =  n0 I  n    . Prove that:
2
2
1)
n0  1
2)
n  1 , and  is a pure state if and only if n  1 .
Prob.
1)
2
2
4. For two qbuts 1 and2 with Pauli matrices  1x ,  1z ,  2 x ,  2 z , prove that
1x 2 x ,1z 2 z   0 , where
2) The eigenvalues of
3) Find
out
the
1x 2 x  1x   2 x
1x 2 x and  1z 2 z are both 1
four
eigenstates
of
 1, 1 ,  1, 1 ,  1, 1 ,  1, 1 ,
states.
1x 2 x
and
 1z 2 z
with
eigenvalues
respectively. These four states are called Bell