summation - How to compute the partial trace of a 4x4 matrix?

I want to compute a partial trace using formula

$\rho_A=\sum\langle B|\rho_{AB}|B\rangle$ .

Example,

$\rho_{AB}= $$ \begin{pmatrix} a & b & c & d\\ e & f & g & h\\ i & j & k & l\\ m & n & o & p \end{pmatrix} $

I try to evaluate

MatrixForm[
 Sum[
   TensorProduct[{{a, b, c, d}, {e, f, g, h}, {i, j, k, l}, {m, n, o, p}}, (q*r)], 

   {1, 2}]]

but Mathematica returns the expression unevaluated.

The output is wrong. I am not sure whether the operator should be TensorProduct or Cross.

I expected to get

$\rho_{A}= $$ \begin{pmatrix} a +f & c+h\\ i+n & k+p\\ \end{pmatrix} $

Answer

Your example can be achieved using Map with a level specification, Partition to generate the sub-matrices and Tr to calculate the traces.

ClearAll[a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p]
r = {{a, b, c, d}, {e, f, g, h}, {i, j, k, l}, {m, n, o, p}};


Map[Tr, Partition[r, {2, 2}], {2}]

{{a + f, c + h}, {i + n, k + p}}