I What does simplicial mean in the context of QM?

What do they mean outside quantum mechanics and what makes you think they mean anything within quantum mechanics ?

 

http://arxiv.org/pdf/1112.2347.pdf

 

Another word for simplicial is polytope - it's a shape with pointy points like a square or a triangle. In a classical probabilistic theory, every mixture is a unique mix of pure states. But this is not true for a mixed density matrix, which can be written in many ways as a mixture of pure density matrices.

 

Usually the quantum set is pictured as a circle.