On the pseudo-manifold of quantum states

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.

arXiv comment18 pages, 4 figures. Minor corrections

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Cited by in corpus (3)

• Parametric models and information geometry on W*-algebras
• Group actions and Monotone Quantum Metric Tensors
• G-dual teleparallel connections in Information Geometry