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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?

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  • $\begingroup$ You have to fix the order in which you apply the unitary operators! $\endgroup$ – Norbert Schuch Jun 8 at 10:56
  • $\begingroup$ 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. $\endgroup$ – lcv Jun 8 at 13:23
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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, $U'_{ABC}\neq U_{ABC}$.

It is actually the same in the bipartite case, where you have

$$ U_{AB}=(U_A\otimes \mathbb 1_B)(\mathbb 1_A\otimes U_B)$$ only in this case the two terms commute and you can simply write

$$ U_{AB}=U_A\otimes U_B.$$

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