linear algebra – Perform a simple operation with operators

I have a map set as

$ Phi (X) = a ^ 2 Tr[X] | 0 rangle langle 0 | + b ^ 2 Tr[X] | 1 rangle langle 1 | + ab Tr[sigma_z X] | 0 rangle langle 1 | + ab Tr[sigma_z X] | 1 row langle 0 |.
$

Right here $ | 0 rangle = begin {pmatrix} 1 \ 0 end {pmatrix} $ and $ | 1 rangle = begin {pmatrix} 0 \ 1 end {pmatrix} $. I want to perform the following operation

$ sum limits_ {j = 0} ^ {1} sum limits_ {k = 0} ^ {1} Psi ( j rangle langle k |) otimes | j rangle langle k |. $

How can this operation be done on Mathematica?