# Intuitive meaning of the eigenvalues and eigenvectors of a reduced density matrix

I am trying to figure out the intuitive meaning behind the eigenvalues and eigenvectors of a reduced density matrix. I understand that of a density matrix- the eigenvalues are the probability the state will be measured in the corresponding pure state eigenvector. What about a reduced density matrix?

I am trying to figure out why all eigenvalues of a reduced density matrix having two-fold degeneracy is a sufficient condition for entanglement. Any insight here would be appreciated.

The interesting quantity of a density matrix (reduced or not) is its rank, that is, the number of non-zero eigenvalues it has. Rank 1 means the state is pure which can also be seen from the density matrix that takes the form of a projection on a single state $$\rho=|\psi\rangle\langle\psi|.$$
Higher rank density matrix implies the state is mixed. The eigenvectors associated with the different eigenvalues $$p_1,p_2,...$$ are the pure states in this mixture $$\rho=p_1|\psi_1\rangle\langle\psi_1|+p_2|\psi_2\rangle\langle\psi_2|+\ldots$$