  We investigate the necessary and sufficient condition for a convex cone of
positive semidefinite operators to be fixed by a unital quantum operation
$\phi$ acting on finite-dimensional quantum states. By reducing this problem to
the problem of simultaneous diagonalization of the Kraus operators associated
with $\phi$, we can completely characterize the kind of quantum states that are
fixed by $\phi$. Our work has several applications. It gives a simple proof of
the structural characterization of a unital quantum operation that acts on
finite-dimensional quantum states --- a result not explicitly mentioned in
earlier studies. It also provides a necessary and sufficient condition for what
kind of measurement statistics is preserved by a unital quantum operation.
Finally, our result clarifies and extends the work of St{\o}rmer by giving a
proof of a reduction theorem on the unassisted and entanglement-assisted
classical capacities, coherent information, and minimal output Renyi entropy of
a unital channel acting on finite-dimensional quantum state.
