  In this article, we study the problem of comparing mixed quantum states:
given $n$ unknown mixed quantum states, can one determine whether they are
identical or not with an unambiguous quantum measurement? We first study
universal comparison of mixed quantum states, and prove that this task is
generally impossible to accomplish. Then, we focus on unambiguous comparison of
$n$ mixed quantum states arbitrarily chosen from a set of $k$ mixed quantum
states. The condition for the existence of an unambiguous measurement operator
which can produce a conclusive result when the unknown states are actually the
same and the condition for the existence of an unambiguous measurement operator
when the unknown states are actually different are studied independently. We
derive a necessary and sufficient condition for the existence of the first
measurement operator, and a necessary condition and two sufficient conditions
for the second. Furthermore, we find that the sufficiency of the necessary
condition for the second measurement operator has simple and interesting
dependence on $n$ and $k$. At the end, a unified condition is obtained for the
simultaneous existence of these two unambiguous measurement operators.
