Geometrical structures for classical and quantum probability spaces

Abstract

On the affine space containing the space S of quantum states of finitedimensional systems there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding relevant geometrical properties of S. Guided by Dirac's analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyse their geometrical properties. Most of our considerations will be dealt with for the simple example of a threelevel system.

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

“…We will now exploit the Jordan–Lie-algebra structure of introduced above to obtain geometric tensor fields on , specifically, we obtain a symmetric, contravariant bivector field associated with the Jordan product , and a Poisson bivector field associated with the Lie product on . This is the generalization to a generic (finite-dimensional) -algebra of what is done in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ] for the specific case for a finite-dimensional Hilbert space . Then, we will show how the manifolds of positive linear functionals introduced in the previous section may be interpreted as a sort of analogs of symplectic leaves for the symmetric tensor in a sense that will be specified later.…”

“…If for some finite-dimensional Hilbert space , it is a matter of direct inspection to show that tensor fields and defined above coincide with those introduced in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ].…”

“…From a purely theoretical point of view, there is no need to restrict our attention to geometrical structures on quantum states associated with covariant tensor fields as in the case of metric tensors discussed above. Indeed, in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ], the associative product of the algebra of linear operators on the finite-dimensional Hilbert space associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on , 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 ]). These two tensor fields, named and , are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in , respectively.…”

The quantum geometric tensor from generating functions

“…This metric in a mathematical context corresponds to the Fubini-Study metric [4,5], and it has been extensively studied in the field of quantum information theory, see for instance [6,7]. Furthermore, in recent years, a lot of work has been done in order to understand the role that information theory plays in Quantum Mechanics, and it has been found that within this theory there is a rich geometric structure [3,8,9,10,11], where play role several structures. One of these approaches consists in defining the Quantum Geometric Tensor (QGT) of the parameter space of the system [3].…”

Non-monotone metric on the quantum parametric model

“…Information geometry on the quantum state space is of great importance from geometrical understanding of quantum theory as well as pure mathematical interest [1][2][3][4][5][6][7][8]. Indeed, the success of geometrical description of quantum theory in various problems should be stressed [9][10][11][12][13][14][15][16].…”