# Heisenberg equation of motion in Liouville space

Consider a quantum system with finite dimensional Hilbert space. Assume that on this Hilbert space, there exist a basis of operators $$G_i$$, so that any aribtrary operator $$A$$ can be expressed as a linear comination of the $$G_i$$ according to $$A=\sum_ia_iG_i$$, where the coefficients $$a_i$$ can be obtained by the Hilbert-Schmidt scalar product according to $$a_i=\text{Tr}(G_i^\dagger A)$$. Furthermore we assume $$\left[G_i,G_j\right]=\sum_kc_k^{ij}G_k$$. The space spanned by the operators $$G_i$$ is often called Liouville space.

Let us now consider some arbitrary operator $$V(t)=\sum_iv_i(t)G_i$$ with explicit time dependence in the Schrödinger picture. Let us furthermore assume that the dynamics of the systen is governed by a time-dependent Hamiltonian $$H(t)=\sum_ih_i(t)G_i$$ generating the time evolution operator $$U(t)$$. Then $$V(t)$$ in the Heisenberg picture takes the form $$\tilde{V}(t)=U^\dagger(t)V(t)U(t)=\sum_i\tilde{v}_i(t)G_i$$. Hereby the tilde denotes the Heisenberg picture. Obviously, $$\tilde{V}(t)$$ fulfills the Heisenberg equation of motion $$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\tilde{V}(t)=\frac{\mathrm{i}}{\hbar}\left[\tilde{H}(t),\tilde{V}(t)\right]+U^\dagger(t)\left(\frac{\mathrm{d}}{\mathrm{d}t}V(t)\right)U^\dagger(t). \label{eq:HE} \end{equation}$$ My question is how I can find the Liouville representation of the Heisenberg equation of motion? To my understanding this would be a differential equation for $$v_i(t)$$ or $$\tilde{v}_i(t)$$, arising from the Heisenberg equation of motion. According to my calculations, as long, as $$V(t)$$ depends explicitely on time, on cannot find an equation depending either only on $$v_i(t)$$ or only on $$\tilde{v}_i(t)$$. Is that correct?

In this paper, in Appendix A, it is said the Liouville representation of the Heisenberg equation is given by $$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\vec{v}(t)=\left(\frac{\mathrm{i}}{\hbar}\left[H(t),\cdot\right]+\frac{\partial}{\partial t}\right)\vec{v}(t). \end{equation}$$ I cannot really make any sense of this equation and I am not able to derive it from the basic assumptions given here despite several trials. Might it be that this equation is wrong? Any help would be really appreciated.

It seems to me that the authors, instead of keeping the basis $$G_i$$ and changing the decomposition vector $$v_i(t) \rightarrow \tilde v_i(t)$$, change the decomposition basis
$$\tilde V(t) = \sum\limits_i \tilde v_i(t) G_i = \sum\limits_i v_i(t) \tilde G_i(t)$$.
The operators $$\tilde G_i(t)$$ still build a basis, as the transformation $$U(t)$$ is unitary. From the evolution equation for $$\tilde V(t)$$ in Heisenberg picture one can derive then equation which confused you.