Density operators as an arena for differential geometry

Vladimir I. Man’ko, Giuseppe Marmo, F. Zaccaria +1 more

Sam Cree, Jonathan Sorce

Miftachul Hadi

We study the algebraic structure of a spinor differential operator associated with the Berry connection, showing that it is a rank-one operator with a simple multiplication law. We establish its relation to the Berry projector through its anti-Hermitian part. These results provide an operator framework for Berry geometry and a basis for future extensions.

Jan Naudts

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.

Apoorva G. Wagh, Veer Chand Rakhecha

Ravinder Rupchand Puri