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I have a map defined as

$\qquad \Phi(X) = a^2\, Tr[X] |0\rangle \langle 0| + b^2\, Tr[X] |1\rangle \langle 1| + a\,b\, Tr[\sigma_z X] |0\rangle \langle 1| + a\,b\, Tr[\sigma_z X] |1\rangle \langle 0| $

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?

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closed as off-topic by Henrik Schumacher, MarcoB, Alex Trounev, eyorble, rhermans Apr 11 at 9:11

This question appears to be off-topic. The users who voted to close gave these specific reasons:

  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – MarcoB, rhermans
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    $\begingroup$ Tr, Dot, KroneckerProduct (or TensorProduct), Sum. $\endgroup$ – Henrik Schumacher Mar 29 at 7:27
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I am assuming $\Psi = \Phi$ in your question. This is a program that would do that, with self-explanatory names.

ket[0] = {1, 0};
ket[1] = {0, 1};

ketbra[i_, j_] := KroneckerProduct[ket[i], ket[j]]

σz = PauliMatrix[3];

Phi[X_]:= a^2 Tr[X] ketbra[0,0] + b^2 Tr[X] ketbra[1,1] + a b Tr[σz.X] (ketbra[1,0] + ketbra[0,1])

And then compute

In[1]:= Sum[KroneckerProduct[Phi[ketbra[j, k]], ketbra[j, k]], {j, 0, 1}, {k, 0, 1}];
        %//MatrixForm

Out[1]//MatrixForm= $$\left( \begin{array}{cccc} a^2 & 0 & a b & 0 \\ 0 & a^2 & 0 & -a b \\ a b & 0 & b^2 & 0 \\ 0 & -a b & 0 & b^2 \\ \end{array} \right)$$

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