Abstract
The Jordan product on the self-adjoint part of a finite-dimensional -algebra is shown to give rise to Riemannian metric tensors on suitable manifolds of states on , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra of linear operators on a finite-dimensional Hilbert space , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.
Highlights
The study of geometrical structures on the space of classical and quantum states is a well-developed and constantly growing subject
In the context of quantum information theory, it is well-known that there is an infinite number of metric tensors on the manifold of faithful quantum states satisfying a property, which is the quantum analog of the monotonicity under Markov maps characterizing the Fisher–Rao metric tensor in the classical case
We presented a geometrical description of one of these quantum metric tensors, the so-called Bures–Helstrom metric tensor, which is rooted in the C ∗ -algebraic nature of the space of quantum states
Summary
The study of geometrical structures on the space of classical and quantum states is a well-developed and constantly growing subject. In [19,20,21,22,23,24,25], the associative product of the algebra B(H) of linear operators on the finite-dimensional Hilbert space H associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on H, and these tensor fields have been used to give a geometrical description of the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation describing the dynamical evolution of open quantum systems (see [19,21,22,24,26,27,28]).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
