Can mixed states be on the boundary of the set of states?

Question

I am confused over the behaviour of states that occupy the boundary of a given set of states. Can they be only pure states, or some mixed states too can occupy the boundary?

Answer 1

The answer to your question depends on what exactly you mean by "boundary": there is the notion of extreme points and the notion of the topological boundary, i.e. points which in their neighbourhood have points from inside as well as outside the original set.

To illustrate: in the case of a triangle the extreme points are the three corners while the topological boundary also includes the three sides. Now for the set of quantum states, the extreme points are the pure states $|\psi\rangle\langle\psi|$ while the topological boundary (with respect to the set of all Hermitian matrices) is the set of all quantum states which have at least one zero eigenvalue. In particular, there are mixed states in the boundary as long as they are not invertible.