# Extract the dynamical map from the density matrix

I am working on open quantum systems and have been wondering whether there is a way to extract the dynamical map using mathematica. Say we have a density matrix

$$\rho(t) = \begin{bmatrix} p_{00} + (1 - |A(t)|^2) p_{11} & A(t)^* p_{01} \\ A(t) p_{01}^* & |A(t)|^2 p_{11} \end{bmatrix}$$

Which can be rewritten as:

$$\rho(t) = \phi(t)\rho(0)=\begin{bmatrix} 1 & 0 & 0& 1-|A(t)|^2\\ 0 & A(t)^* & 0 & 0\\ 0 & 0 & A(t) & 0\\ 0 & 0 & 0 & |A(t)|^2 \end{bmatrix} \begin{bmatrix}\rho_{00}\\\rho_{01}\\\rho_{10}\\\rho_{11}\end{bmatrix}$$

where $$\phi(t)$$ is the dynamical map. Here we write the density matrix at time t but as a vector which can be deconstructed into a matrix applied to a vector of the elements of the density matrix at the initial state. For simple systems it is doable by hand, however for higher level systems it becomes extremely tedious.

For non-physicists, density matrices always have trace 1, are hermitian and are positive semi-definite. I wish to extract the dynamical map symbolically. I think this would be similar to extracting a coefficient matrix as in How to read off coefficients of tensor-like expression in a speedy way? and Extract coefficient matrix $A$ from expression $f(x)$.

The dynamical map itself does not depend on the initial state. It is a function which temporally evolves the input state:

$$\rho(t) = \begin{bmatrix} p_{00} & p_{01} \\ p_{01}^* & p_{11} \end{bmatrix}$$

• What is the characterization of $p_{ij}$ and $\rho_{ij}$? Oct 11 '21 at 18:55
• From what I can gather, $\rho_{ij}$ is the $2\times 2$ matrix with $p_{ij}$ in the $ij$ spot and 0's elsewhere, conjugated according to some rules—is that right? If so, the crucial question is: what data do you actually have in mathematica? Is it numerical, or symbolic? Do you know which parts are coefficients and which aren't? For example, why shouldn't $p_{01}$ be absorbed into the definition of $\phi(t)$ as a coefficient on $A(t)^*$? Is there a standard form $\phi$ is assumed to have in this context? I think this question should be edited to include more detail, but it sounds interesting! Oct 11 '21 at 19:37
• Density matrices always have trace 1, are hermitian and are positive semi-definite. I wish to extract the dynamical map symbolically. I think this would be similar to extracting a coefficient matrix as in mathematica.stackexchange.com/questions/23039/… and mathematica.stackexchange.com/questions/83496/….
– dan
Oct 12 '21 at 17:03
• I think my edits should answer your question. The dynamical map is sort of like a matrix of coefficients to the initial state elements. The initial state elements are the variables.
– dan
Oct 12 '21 at 17:28
• dan, you should first establish how exactly the square matrices should be formed into a list/column/row vector. It seems this would be a crucial step here, no? Oct 13 '21 at 8:18