I am confused about what it is exactly that a reduced density operator describes. To illustrate, I came across the following seemingly paradoxical argument.
Consider a bipartite system $AB$, described by the pure state:
$$| \psi \rangle = \sum_{a,b} \psi(a,b) |ab \rangle$$
Its density matrix is defined as:
$$\rho \equiv | \psi \rangle \langle \psi |$$
With reduced density operators:
$$\rho_A = tr_B(\rho) \qquad \rho_B = tr_A(\rho)$$
We can construct a new operator:
$$\rho_{AB} = \rho_A \otimes \rho_B$$
Which is only equal to $\rho$ if $|\psi\rangle$ is a separable state (no correlations between $A$ and $B$). Another way to say this is:
$$0 = S(\rho) \leq S(\rho_A) + S(\rho_B) = S(\rho_A \otimes \rho_B) = S(\rho_{AB})$$
Because if $A$ and $B$ are entangled, then $\rho$ contains information about these correlations, but $\rho_{AB}$ does not (is this wrong?).
So say $A$ and $B$ are entangled, then $\rho_{AB}$ does not contain that information, but I was under the impression that $\rho_A$ ánd $\rho_B$ do contain (some, all?) of this correlation information; otherwise both $A$ and $B$ should be described by pure states. I realise that $\rho_{AB}$ has no physical meaning in the case of entanglement, but even as a purely mathematical construct I don't see how that information can just vanish.
I guess my question is this: what information exactly does the reduced density operator encompass? And, if any of it is related to the correlations, how can this be reconciled with the argument in the paragraph above?
