Suppose have a bosonic state as follow:
$a \left| 0nm \right> \left< 0nm \right|+b \left| 0nm \right> \left< 1(n+1)(m+1) \right|+c \left| 1(n+1)(m+1)\right> \left< 0nm \right|+d \left| 1(n+1)(m+1)\right> \left< 1(n+1)(m+1)\right| $
where n and m are 0,1,2,...
also 0 and 1 are basis of a qubit.
its not possible to represent the density matrix of this state by basis 1 , 2 , ....
we have to use this basis instead to represent the state
$\left| 0nm\right>,\left| 0(n+1)m\right>,\left| 0(n+1)(m+1)\right>,\left| 0n(m+1)\right> \left| 1nm\right>,\left| 1(n+1)m\right>,\left| 1(n+1)(m+1)\right>,\left| 1n(m+1)\right>$
if we do that can we calculate eigenvalues of the matrix like a normal finite matrix?
if not then how we can do that?
