Hamilton-Jacobi Theory and Information Geometry

Recently, a method to dynamically define a divergence function $D$ for a given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function $\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function $D$ to be known and we look for a Lagrangian function $\mathfrak{L}$ for which $D$ is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.Comment: 8 page

Lagrangian description of Heisenberg and Landau-von Neumann equations of motion

An explicit Lagrangian description is given for the Heisenberg equation on the algebra of operators of a quantum system, and for the Landau-von Neumann equation on the manifold of quantum states which are isospectral with respect to a fixed reference quantum state.Comment: 13 page

Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems

In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. In order to do so, we construct a Hamilton–Jacobi theory for these systems following two approaches, obtaining two different Hamilton–Jacobi equations. Compared to conservative Hamiltonian systems, contact Hamiltonian systems depend of one additional parameter. The fact of obtaining two equations reflects whether we are looking for solutions depending on this additional parameter or not. In order to illustrate the theory developed in this paper, we study three examples: the free particle with a linear external force, the freely falling particle with linear dissipation and the damped and forced harmonic oscillator

The space of Quantum States, a Differential Geometric Setting

The subject of this thesis is the geometry of the space of quantum states. The aim of this thesis is to present a geometrical analysis of the structural properties of this space, being them of ``kinematical'' or ``dynamical'' character. We will see that the space of quantum states of finite-dimensional systems may be partitioned into the union of disjoint orbits of the complexification of the unitary group. These orbits are the manifolds of quantum states with fixed rank. On the one hand, we will compute the two-parameter family of quantum metric tensors associated with the two-parameter family of quantum q-z-Rényi relative entropies on the manifold of invertible quantum states (maximal rank). Using the powerful language of differential geometry we are able to perform all the computations in an arbitrary number of (finite) dimensions without the need to introduce explicit coordinate systems. On the other hand, we will develop a geometrization of the GKLS equation for the dynamical evolution of Markovian open quantum systems. Specifically, we will write the GKLS generator by means of an affine vector field on an affine space, and we will decompose this vector field into the sum of a Hamiltonian vector field, a gradient-like vector field, and a so-called Kraus vector field. This geometrization will be used in order to analyze and completely characterize the asymptotic behaviour of the dynamical evolutions known as quantum random unitary semigroups by means of the so-called purity function. Finally, we will comment on the possibility of extending the results presented to the infinite-dimensional case, and to the case of multipartite quantum systems