A doubt regarding Black Hole Complementarity

Suppose you have two systems $S_1,S_2$ with Hilbert spaces $H_1,H_2$ with a density matrix $\rho$ on $H_1\otimes H_2$. The partial trace of $\rho$ over the Hilbert space of one of the systems, $H_1$ say gives you a reduced density matrix $\rho_2$. The reduced density matrix predicts the expectation values of all the measurements you can conduct on $S_2$ alone. It does not predict expectation values of measurements on $S_1$, or the expectation values of measurements on the joint system.

In this case, the universe outside the black hole supposedly loses access to information about what falls into the black hole. So the information available to those outside the black hole is described by a reduced density matrix. Whether such loss actually happens is unknown.