I was going through the book "The theory of open quantum systems" by Breuer and Petruccione, and I am having problems with convincing myself of equation 3.49. In short, I am reading about Quantum Markovian Processes and Markovian master equations. The system at hand is a joint system formed by a open system $\rho_S$ coupled to an environment described by a density operator $\rho_E$. The following is the equation I am having issues with
$$ W_{ab}(t) = \sum_i^{N^2}F_i(F_i,W_{ab}(t)) \;\;\; (3.49)$$ $$ for\;\;(F_i,F_j)\equiv tr_S(F_i^\dagger F_j)=\delta_{ij} $$
where $F_i$ ($i=1,2,…,N^2)$ form a complete basis of orthonormal operators in the Liouville space corresponding to the Hilbert space $H_S$ ($dim(H_S)=N$) of an open quantum system $\rho_S$. Moreover $F_{N²}=\sqrt{1/N}I_S $, so that $tr_S(F_j)=0$ for $j=1,…,N²-1$.Furthermore we have
$$W_{ab} = \sum_{ab}\sqrt{\lambda_b} \langle\phi_a|U(t,0)|\phi_b\rangle,$$
where $\rho_E = \sum_a \lambda_a |\phi_a\rangle\langle\phi_a|$ is the spectral decomposition of the environment in the joint system $\rho(0) = \rho_S(0) \otimes \rho_E$; where, lastly, the unitary $U(t,0)$ determines the evolution of the total system
$$\rho(t) = U(t,0) (\rho_S(0) \otimes \rho_E) U^\dagger(t,0). $$
For some more context, the operator $W_{ab}$ is introduced in the book to give a representation to the dynamical map $V(t): S(H_S) \rightarrow S(H_S)$, where $$V(t)\rho_S(0)=\rho_S(t) = tr_E (U(t,0) (\rho_S(0) \otimes \rho_E) U^\dagger(t,0)).$$ The way the operator $W_{ab}$ is introduced is by then considering the spectral decomposition of $\rho_E$ which inserted in the expression for $\rho_S(t)$ gives the form of $W_{ab}$ I provided above: which ultimately leads to
$$\rho_S(t) = \sum_{ab} W_{ab} \rho_S(0) W^\dagger_{ab}$$
In synthesis the question is: how am I supposed to interpret the product between $F_i$ and $W_{ab}$ given that they represent operators acting on different spaces with different dimensionality?
Best Answer
They do act on the same space: The $\{F_i\}$ are basis of the Hilbert space of operators acting on $H_S$. The $W_{ab}$ are operators acting on $H_S$: They came from the unitary operator on the whole system + environment Hilbert space, say $H_{SE}$, but after tracing out the environmental degrees of freedom, they act only on $H_S$. The statement of Eq. 3.49 is just a completeness relation for operators in Hilbert space $H_S$.
Seeing the comments, maybe it would also be important to clarify that $N^2$ is not the dimension of the space where the $\{F_i\}$ act on, but rather the number of elements in $\{F_i\}$. This means that the dimension of the Hilbert space of operators (also called Banach space) is of dimension $N^2$.