Density matrix exponentiation now allows us to apply the quantum phase algorithm to find the eigenvectors and eigenvalues of an unknown density matrix.If we have n copies of p,we use the ability to apply e to perform the quantum phase algorithm'. In particular,the quantum phase algorithm uses conditional applications ofei for varying times t to take any initial state)0) to∑；y;lXi)li),where|Xi〉are the eigenvectors of p,;are estimates of the corresponding eigenvalues,and v;=(xilv)