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-Renyi 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.
We review the geometry of the space of quantum states
S(H) of a finite-level quantum system with Hilbert space
H from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group
SU(H) and its complexification
SL(H), the complex special linear group. A stratum of the stratification generated by
SU(H) is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by
SL(H) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of
SL(H) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this
SL(H)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on
S(H) that preserve the von Neumann entropy and the convex structure of
S(H), while the latter give rise to invertible maps on
S(H) that preserve the von Neumann entropy but not the convex structure of
S(H). A similar decomposition is given for the
SL(H)-action generating the stratification of
S(H) into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition.
Dynamical evolution is described as a parallel section on an infinite-dimensional Hilbert bundle over the base manifold of all frames of reference. The parallel section is defined by an operator-valued connection whose components are the generators of the relativity group acting on the base manifold. In the case of Galilean transformations we show that the property that the curvature for the fundamental connection must be zero is just the Heisenberg equations of motion and the canonical commutation relation in geometric language. We then consider linear and circular accelerating frames and show that pseudoforces must appear naturally in the Hamiltonian.
Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group
ϕt of transformations on a Hilbert space,
H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow
ϕt. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow
ϕt. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.