# Research

## Research lines

At present, my main research lines are the following:

### The geometrical formulation of Quantum Mechanics

Classical mechanics (CM) can be formulated in several mathematical frameworks each corresponding to a different level of abstraction: Newton equations, the Hamiltonian formalism, the Poisson brackets, etc. Perhaps its more abstract and general formulation is geometrical, in terms of Dirac manifolds. Similarly, Quantum Mechanics (QM) can also be formulated in different ways, some of which resemble its classical counterpart.

The space of physical states are considered as a real differentiable manifold \$M_Q\$ which is obtained by realifying the complex vector space which is the carrier space of the Hilbert space cotaining the states of the quantum system. The scalar product of the Hilbert space is encoded in three tensors defined on the real vector space \$M_Q\$. Two of these tensors correspond to a a metric tensor and a symplectic one which allow us to write the expression of the Schrödinger equation as a Hamilton equation, in a form which is completely analogous to the Hamiltonian formulation of Classical Mechanics. The third one is a complex structure tensor on \$M_Q\$ that, together the two previous tensors, provide \$M_Q\$ with a Kähler structure.

Physical observables (self--adjoint linear operators) are considered as real quadratic functions defined on the manifold \$M_Q\$, corresponding to the average value that the given observable takes on each point of \$M_Q\$. The set of functions can be endowed with a Poisson algebra structure almost equal to the one that characterizes the dynamical variables in CM. Moreover, Schrödinger equation can be recast into Hamiltonian equations form by transforming the complex Hilbert space into a real one of double dimension. The observables are also transformed into dynamical functions in this new phase space, in analogy to the classical one. Finally, a Poisson bracket formulation has also been established for QM, which permits to classify both the classical and the quantum dynamics under the same heading.

Despite its validity for general quantum systems, it has been applied mostly to finite dimensional Hilbert spaces such as those arising in Quantum Information Theory, in particular the characterization of entanglement in geometrical terms.

Nonetheless, the description provides a nice geometrical framework which can be used in different directions, as for instance:
• #### The description of mixed Quantum-Classical systems

The geometrical description of Quantum Mechanics allows us to combine the dynamical description of the classical and the quantum subsystems as we do when describing a composed classical system: the space of states corresponds to the cartesian product of the factors and the symplectic forms are combined in the natural way. Within this framework, it is simple to prove that Ehrenfest Dynamics is Hamiltonian.

It is also possible to construct a Statistical description of this type of system in a rigorous way, by using the volume form associated to the composed symplectic form. Other properties of the system, such as Nosé thermostats can also be adapted from the classical domain.

• #### Geometrical description of Quantum Statistical Mechanics

Once we have been able to represent in a geometrical language the basic ingredients of Quantum Mechanics and we have seen that the formalism resembles the classical one, we can try to formulate in this framework an ensemble of quantum systems and try to relate it with the analogue description of Classical Statistical Mechanics.

• #### Open quantum systems and decoherence

We can also consider the description of system coupled to a second system, the environment, and use the geometrical tools at our disposal to describe in a simple way the dynamics of the system once we trace out the environment degrees of freedom.

### Optimal control theory on Lie groups

• #### Classical control

In this context we have been working recently in the description of optimal control problems of systems with constraints, and more specifically the use of quasi-coordinates in systems defined on Lie algebroids. This offers a fairly general framework which admits several interesting problems.

One of them, the Riemannian cubic polynomials (RCP) problem, also called Riemannian cubics, can be seen as a generalization of cubic splines in Euclidean space to Riemannian manifolds. When the system is defined on a Lie group, pre-symplectic or symplectic formulations of Pontryagin Maximum Principle allow us to perform a study of the dynamics from the point of view of reduction and integrability threory which highly simplifies the problem.

• #### Quantum control

The geometrical description of Quantum Mechanics allows us also to formulate Quantum Control problem in a common framework with respect to geometric classical control problems. We can study, for instance, optimal control problems where the cost functional is the decoherence, the entanglement between some qubits, etc.

## Student Supervision

#### PhD Students

Lígia Raquel dos Santos Abrunheiro, Universidade de Coimbra (Portugal), 14 September 2012
Co-Supervision: Dra. Margarida Camarinha, Universidade de Coimbra (Portugal)
Thesis : Polinomios cubicos riemannianos: uma abordagem hamiltoniana e aplicaçoes

Jorge A. Jover-Galtier, Universidad de Zaragoza, 3 July 2017
Co-Supervision: Prof. José F. Cariñena Marzo
Thesis : Open quantum systems: geometric description, dynamics and control
Grade : Sobresaliente cum laude (Highest possible)

#### Master Students

Teresa Ubieto Puértolas, Universidad de Zaragoza, June 2012
Co-Supervision: Dr. Jacobo Cano Escoriaza
MSc Thesis : Desarrollo de herramientas didácticas para la plataforma Ibercivis

Jorge Alberto Jover Galtier, Universidad de Zaragoza, 10 July 2013
MSc Thesis : Fenómenos de decoherencia en sistemas moleculares

Jorge Alberto Jover Galtier, Universidad de Zaragoza, 9 July 2012
TAD Thesis : Geometric Formulation of Quantum Mechanics

Diego Medrano Jiménez, Universidad de Zaragoza, 26 September 2014
Degree Thesis in Physics: Formulación geométrica de la mecánica cuántica y sus aplicaciones
Grade : Matrícula de honor (highest possible)

Pablo Sala de Torres-Solanot, Universidad de Zaragoza, 9 June 2015
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Mathematics : Formulación simpléctica de la mecánica cuántica
Grade : Matrícula de honor (highest possible)

Degree Thesis in Physics : Formulación geométrica de la mecánica cuántica y sus aplicaciones
Grade : Matrícula de honor (highest possible)

Alfonso Lanuza García, Universidad de Zaragoza, 8 July 2016
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Mathematics : Formalismo geométrico de la mecánica cuántica. Integradores Unitarios
Grade : Matrícula de honor (highest possible)

Néstor González Gracia, Universidad de Zaragoza, 7 July 2016
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Physics : Aplicaciones de la formulación geométrica de la Mecánica Cuántica. Evolución Markoviana en sistemas cuánticos.