Time evolution of a tripartite quantum state

Question

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?

Answer 1

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

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

Answer 2

If you have a tripartite system and you do want to compute it's evolution, you need to write the global Unitarty Operator, evolve the global system and get the informations of some subsystem, if you want so.

1. Write the Global Unitary over the composed Hilbert Space, i.e., $U_{G}$ acts over $\mathcal{H}_G=\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$, where: $dim(\mathcal{H}_G) = dim(\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C) = d_{A}\cdot d_{B}\cdot d_{C}$ and $U_{G} \in \mathbb{C}^{(d_{A}\cdot d_{B}\cdot d_{C}) \times (d_{A}\cdot d_{B}\cdot d_{C})}$. If we're talking about QuBits: $dim(\mathcal{H}_G) = 2^3$ and $U_{G} \in \mathbb{C}^{2^3 \times 2^3}$.

2. Apply $U_{G}$ over the global state: \begin{align} \rho'_{G} &= U_{G}\rho_{G}U^{\dagger}_{G}\\ & = U_{G}\left[\rho_{A}\otimes\rho_{B}\otimes\rho_{C} \right] U^{\dagger}_{G} \end{align}

3. If you want the information about any subsystem: \begin{align} \rho'_{A} &= tr_{BC}\left[U_{G}\left(\rho_{A}\otimes\rho_{B}\otimes\rho_{C} \right) U^{\dagger}_{G}\right]\\ \rho'_{B} &= tr_{BC}\left[U_{G}\left(\rho_{A}\otimes\rho_{B}\otimes\rho_{C} \right) U^{\dagger}_{G}\right]\\ \rho'_{C} &= tr_{BC}\left[U_{G}\left(\rho_{A}\otimes\rho_{B}\otimes\rho_{C} \right) U^{\dagger}_{G}\right] \end{align}

However, I think $U_{G} = U_{A} \otimes U_{B} \otimes U_{C}$ doesn't acts globally, because if $U_{G}= U_{A} \otimes U_{B} \otimes U_{C}$, then: \begin{align} \rho'_{G} &= U_{G}\rho_{G}U^{\dagger}_{G}\\ & = U_{G}\left[\rho_{A}\otimes\rho_{B}\otimes\rho_{C} \right] U^{\dagger}_{G}\\ & = U_{A}\rho_{A}\otimes U_{B}\rho_{B}\otimes U_{C}\rho_{C} \end{align} which is a separable system. Typically, Global Unitaries entangle the subsystems.

To construct $U_{G}$ properly, I think you can:

• Build it using ideas of Kraus Formalism for Open Quantum Systems which you may determine one subsystem as a ancilla subsystem.
• Find the Global Hamiltonian Operator $\hat{H}_{G}=\hat{H}_{ABC}$ and $U_{G}=e^{-iH_{G}t/\hbar}$ follows directly.