# [Physics] Importance of zero and non-zero eigenvalues of density matrix

density-operatorquantum mechanicsquantum-information

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?

What can we say about the quantum state from the number of zero and non-zero eigenvalues of the corresponding density matrix?

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.

For practical computations, one can just decompose the density matrix into a sum of pure states. The optimal way to do this (i.e. that you get the least error for the number of pure states that you use) is the optimal coherent decomposition, where you compute the eigenvalue decomposition of the density matrix. The dynamics of SchrÃ¶dinger equations is such that any such decomposition stays valid (and optimal) during time propagation, i.e. you can just propagate each individual pure state.

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.

Anything related to entanglement or any other properties? Does they vary with the nature of states such as it is pure or mixed?

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.