Are the pure quantum states that make up a mixed state orthogonal?

Question

I came a cross a problem when leaning Quantum PCA. I know that a mixed state is an ensemble of several quantum states. But I wonder whether these states that make up for the mixed state are orthogonal. If not, under what condition they are? I'm glad if you could help me with this:)

Answer 1

The decomposition of a mixed state is not unique. So, in general, there are both orthogonal and non-orthogonal decompositions of the same state. However, note that there is always an orthogonal decomposition, just made up of a mixture of the eigenvectors of the density matrix. (As the density matrix is Hermitian, its eigenvectors are always orthogonal).