What can we say about the quantum state from the number of zero and non-zero eigenvalues of the corresponding density matrix? Anything related to entanglement or any other properties? Does they vary with the nature of states such as it is pure or mixed?

Please add some references.

## Best Answer

The number of zero eigenvalues has no significance, and is not really well defined anyway.

If the number of non-zero eigenvalues is not one, then there are many different ways to write the density matrix $\rho$ as a coherent decompositions of the form $\rho = \sum_k p_k|\psi_k\rangle\langle\psi_k|$ with $\langle\psi_k|\psi_k\rangle=1$ and $p_i\geq p_j \geq 0$ for $i \leq j$. Iff $\langle\psi_i|\psi_j\rangle=\delta_{ij}$, then this decomposition is an eigendecomposition. Because $\rho$ is Hermitian and positive, an eigendecomposition is also a singular value decomposition, and hence describes all optimal low rank approximations (with respect to the Euclidean norm) in a succinct form. Hence this decomposition is somtimes called optimal coherent decomposition by some communities.

More pragmatically, I recently explained this as follows:

The last sentence of this pragmatic explanation assumes that $\langle\psi_i(t)|\psi_j(t)\rangle=\langle\psi_i(t_0)|\psi_j(t_0)\rangle$ is preserved during time propagation, which is valid for "closed" systems.

As others pointed out, an entangled state is also a pure state. If you compute a partial trace over an entangled state, you get a mixed state, but this is not really related to the eigendecomposition. But this is an interesting observation nevertheless, because the optimal coherent decomposition for the corresponding subsystem won't be preserved in general during time propagation, and hence there can be some sort of quantum leap from the perspective of the subsystem in terms of the optimal coherent decomposition. But the optimal coherent decomposition is only unique if $p_i> p_j \geq 0$ for $i < j$ anyway.