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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.

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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 $$\rho=p_1|\psi_1\rangle\langle\psi_1|+p_2|\psi_2\rangle\langle\psi_2|+\ldots$$
(When some of the eigenvalues are degenerate the eigenvectors are somewhat arbitrary but the statement is essentially the same.)

Why multiple non-zero eigenvalues (they don't have to be degenerate) in the reduced state imply entanglement? Because this implies that the reduced state is mixed and (assuming the unreduced state was pure) this can only happen if the unreduced state was entangled. If it is not clear why it is the case, I encourage you to learn about the Schmidt decomposition.

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