What information can we get out about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix?
For example, it can be proved that if all the eigenvalues of a density matrix are positive, then the eigenvalues of the reduced density matrix are positive too.
From here, it looks like knowing all the eigenvalues is a hard problem, but what information can we possibly obtain? Information about the determinant perhaps?
I cannot answer the question completely. But we can have look at some of the edge cases. For the case of a bipartite pure system with the same number of qubits on both sides, nothing can be said about the eigenvalues of a reduced system from the eigenvalues of the full system. This is because any density matrix can be purified using this setup.
Now on the other hand if the full system is in the maximally mixed state then its reduced density matrices are also maximally mixed. So, in this case, we know all the eigenvalues of the reduced density matrices.
biryanishowed that going from maximally mixed to pure states you go from fully defined eigenvalues of $\rho^A$ to totally undefined eigenvalues of $\rho^A$ (except of course for a few basic requirements on the eigenvalues that are always satisfied). These statements can be seen as consequences of the subadditivity of the entropy: $S(\rho^A)\ge S(\rho)$. I don't know if this is the only thing that can be said about it though, I'll think about it $\endgroup$