Acting with a superoperator to states in qutip

Question

I can generate a random superoperator in qutip using the command rand_super(N) $\mathcal{E}$ where I only need to inser the dimension of the superoperator is acting on, denoted as N. The same superoperator should be able to act on a state, that is a density matrix, NxN $\rho$.

Question: How can I do this using Qutip? Literally, given $\rho$ I can generate a random $\mathcal{E}$, but I couldn't find $\mathcal{E}(\rho)$.

I don't know if this is really a qutip question, maybe I could do this using simply numpy? I also tried to look up for this and qutip only teaches to find properties of the matrices and not how to perform this specific kind of operation.

Answer 1

For an operator $U$ acting on a state $\rho$ as $U\rho$, it's superoperator representation $\mathcal{E}$ would act on the superket representation of the state $\rho$, which is denoted as $|\rho\rangle\rangle$. This superket is just a vector made up of the columns of $\rho$ stacked on each other: $$ \rho = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \rightarrow |\rho\rangle\rangle = \begin{pmatrix} a \\ c \\ b \\ d \end{pmatrix}. $$ To do this in QuTiP, you have

import qutip as qt

N = 3
rho = qt.rand_dm(N) # Random 3x3 density matrix
rho_vec = qt.operator_to_vector(rho)

You can then act on this super ket with the super operator $\mathcal{E}|\rho\rangle\rangle$. Note that for the equivalent operator $U$ acting on the state $\rho$, $U\rho$ is not necessarily equal to $\rho U$. So you have to keep tabs on whether the superoperator is left- or right-multiplied. A nice explanation is given here. In QuTiP, this is handled by calling spre(rho), spost(rho), and sprepost(rho, rho), detailed here.