Question:
How do we write the unitary evolution of a tripartite system in Hilbert Space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ when it is subject to two unitary evolution operators $U_{AB}$ and $U_{BC}$. $U_{AB}$ a unitary operator in $\mathcal{H}_A \otimes \mathcal{H}_B$ acting on the $A$ and $B$ subsystems, and $U_{BC}$ is a unitary operator in $\mathcal{H}_B \otimes \mathcal{H}_C$, acting on the $B$ and $C$ susbsystems?
More details:
If a density operator $\rho_A$ exists in Hilbert space $\mathcal{H}_A$ and undergoes unitary evolution, this can be written
$$\rho_A(t) = U_A(t)\rho_A(0)U_A(t)^\dagger$$
where $U_A(t)$ is a unitary operator in $\mathcal{H}_A$. Similarly, for a density operator in a Hilbert space $\mathcal{H}_B$, we can write the evolution as
$$\rho_B(t) = U_B(t)\rho_B(0)U_B(t)^\dagger$$
where $U_B$ is a unitary operator in $\mathcal{H}_{B}$.
If we the consider the joint Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B $, with $U_A$ acting only on the $A$ subsystem and $U_B$ acting only on the $B$ subsystem, we can write the evolution of the joint state as
$$\rho_{AB}(t) = U_{AB}(t)( \rho_A(0) \otimes\rho_B(0)) U_{AB}^\dagger(t) $$
where $U_{AB} = U_{A}\otimes U_B$. In this case, there was a simple way of writing the joint evolution operator $U_{AB}$, in terms of the unitary evolutions of the subsystems. My question is whether there is also a simple way of writing the joint evolution operator in the tripartite system described at the start.
Thanks in advance.
Edit:
I think that simply writing $U_{ABC} = U_{AB}\otimes U_{BC}$ is not the correct answer, as it would give an operator of the wrong dimension. Am I correct in thinking this?
