# Time evolution of a tripartite quantum state

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.

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?

• You have to fix the order in which you apply the unitary operators! – Norbert Schuch Jun 8 '20 at 10:56
• I would say you need to know what the Hamiltonian is. You only have one Schroedinger equation and only one evolution operator. So the problem does not exist to begin with. – lcv Jun 8 '20 at 13:23

Since you are applying two consecutive evolutions on $$B$$, you have to decide which one to apply first. You have two possibilities, either
$$U_{ABC}=(U_{AB}\otimes \mathbb 1_C)(\mathbb 1_A\otimes U_{BC})$$ or $$U'_{ABC}=(\mathbb 1_A\otimes U_{BC})(U_{AB}\otimes \mathbb 1_C).$$
In general, $$U'_{ABC}\neq U_{ABC}$$.
$$U_{AB}=(U_A\otimes \mathbb 1_B)(\mathbb 1_A\otimes U_B)$$ only in this case the two terms commute and you can simply write
$$U_{AB}=U_A\otimes U_B.$$