On the dynamics of open quantum systems

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?

• the first equation here is just the decomposition of a vector in an orthonormal basis. In this case the "vector" is an operator, and the basis is made of operators, and the inner product is $\langle A,B\rangle=\operatorname{Tr}(A^\dagger B)$, but the rest doesn't change
– glS
Commented Jul 11, 2022 at 14:24
• arent $W_{ab}$ and $F_i$ acting on two separate spaces with different dimensions? How am I supposed to interpret the product between the two?
– Oti
Commented Jul 11, 2022 at 14:28
• surely you can supply appropriate context without copy-pasting. If not why should the community be interested in such a specialized problem? Commented Jul 11, 2022 at 14:33
• @OtiDioti mh no, wait, maybe I see the source of the confusion. Each $F_i$ acts on $H_S$, which has dimension $N$, and so do $W_{ab}$. At the same time, an Hermitian operator on $H_S$ can be thought of as a vector in a real vector space of dimension $N^2$, which is the space of Hermitian operators defined on $H_S$. That's why the basis $\{F_i\}$ contains $N^2$ different elements. But that doesn't change that each operator, being (representable as) an $N\times N$ matrix, acts on $H_S$. You're dealing with two different vector spaces: $H_S$, and $\operatorname{Herm}(H_S)$, do not confuse them
– glS
Commented Jul 11, 2022 at 15:14
• actually, sorry, on a second read, the text is decomposing general (not necessarily Hermitian) operators on $H_S$. So the inner product is defined in the complex $N^2$-dimensional vector space of all linear operators $H_S\to H_S$. The operators $W_{ab}$ are not Hermitian, but nevertheless can be decomposed using the basis of operators $F_i$. These do not need to be Hermitian either (in fact, you can forget about Hermitianity altogether here, it's not particularly relevant, I only brought it up because I'm used to Hermitian bases of ops)
– glS
Commented Jul 11, 2022 at 15:21

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 $$N^2$$.