Extensions of product states

Question

Given a product state $\rho_{AC} = \rho_A\otimes \rho_C$, what can we say about the structure of states $\rho_{ABC}$ that are extensions of $\rho_{A}\otimes \rho_C$? By extension I mean that $\text{tr}_B\rho_{ABC} = \rho_A \otimes \rho_C$, and I am not assuming that $\rho_{ABC}$ is pure. Ideally, I would know the most general form any such extension can take on. Less ambitiously, I'd like to know any general properties such extension states must have.

The closest related work to this question which I have found concerns the structure of states satisfying $I(A:C|B)=0$. Work by Hayden, Jozsa, Petz and Winter (arXiv:quant-ph/0304007) shows that whenever $I(A:C|B)=0$, there's a way to split up the $B$ Hilbert space $\mathcal{H}_B = \oplus_i \mathcal{H}_{b_L^i}\otimes \mathcal{H}_{b_R^i}$ such that

$$\rho_{ABC} = \sum_i q_i \rho_{Ab_L^i}\otimes \rho_{b_R^iC}$$

which in particular gives that $\rho_{AC}$ is separable. They also obtain as a corollary that $\rho_{AC}$ separable implies the existence of an extension such that $I(A:C|B)=0$.

My question is not addressed directly by the Hayden et al. result however as I am interested 1) in product states specifically, and 2) I'd like to know what general form the extensions are constrained to given that AC is product, rather than about the existence of an extension with a certain property.

Answer 1

The general form of all possible extensions is incredibly general. Let $$ \rho_A=\sum_ip_i|\psi_i\rangle\langle\psi_i|,\qquad \rho_C=\sum_jq_j|\phi_j\rangle\langle\phi_i|, $$ but remember that there are generally many choices for such a decomposition. Now, we could create a pure state $$ |\Psi\rangle=\left(\sum_i\sqrt{p_i}|\psi_i\rangle_A|i\rangle_{B_1}\right)\left(\sum_j\sqrt{q_j}|j\rangle_{B_2}|\phi_j\rangle_C\right)|0\rangle_D. $$ (I wonder if there are even more options if there are commensurate sets of values for $\sqrt{p_iq_j}$.) Now, you can have $$ \rho_{ABC}=\text{Tr}_D\left(U_{B_1B_2D}|\Psi\rangle\langle\Psi|U_{B_1B_2D}^\dagger\right), $$ where $U_{B_1B_2D}$ is a unitary acting over the two $B$ parts, $B_1$ and $B_2$, and some auxiliary system $D$.

When $U$ does not act on $D$, the extension is a pure state. For example, if $U$ is the identity, there's a lot of bipartite entanglement between $A$ and $B_1$ and between $C$ and $B_2$, but no entanglement between $A$ and $C$. I assume varying $U$ gives you some control over entropies such as those conditioned on $B$ (but I haven't tried it).

At the other extreme of $U$, you can swap systems $B$ and $D$, which essentially leaves you with a form $\rho_{ABC}=\rho_A\otimes\rho_B\otimes \rho_C$.

I have no idea what implications this has for entropies and the like.