On the pseudo-manifold of quantum states

TL;DR

This paper clarifies the geometric and stratification structures of quantum state spaces, showing that certain natural partitions are not true stratifications and providing explicit descriptions for matrix algebras.

Contribution

It precisely characterizes the stratification of quantum states, distinguishing between orbit decompositions and Whitney stratifications, and describes the pseudo-manifold structure for matrix algebras.

Findings

Orbit partitions are not stratifications.

Whitney stratification exists for fixed-rank matrices.

Explicit pseudo-manifold structure described for full matrix algebras.

Abstract

There are various statements in the physics literature about the stratification of quantum states, for example into orbits of a unitary group, and about generalized differentiable structures on it. Our aim is to clarify and make precise some of these statements. For A an arbitrary finite-dimensional C*-algebra and U(A) the group of unitary elements of A, we observe that the partition of the state space S(A) into U(A) orbits is not a decomposition and that the decomposition into orbit types is not a stratification (its pieces are not manifolds without boundary), while there is a natural Whitney stratification into matrices of fixed rank. For the latter, when A is a full matrix algebra, we give an explicit description of the pseudo-manifold structure (the conical neighborhood around any point). We then make some comments about the infinite-dimensional case.