## Q-Math Seminar

### Sijo K. Joseph (Universidad Rey Juan Carlos)

#### Chaos and Entanglement in Fiber Optics

##### Monday 8th of June 2015, 11:00, UC3M, Seminar Room 2.2D08

Using a quantum chaotic optical fiber, the non-separable entanglement which appears in the classical electromagnetic field is analyzed. In a specially designed optical fiber, we manipulate the non-separable entanglement by changing the optical fiber geometry which is known to generate chaotic behavior. An optical fiber with a cross section that belongs to the family of Robnik chaotic billiards is considered. Quantum chaotic billiard geometry is used to explore the influence of a mechanical modification of the optical fiber cross-sectional geometry on the production of non-separable entanglement within classical fields. In our proposal non-entangled light enters at the input end of the fiber and entangled light propagates out after interacting with a fiber boundary, which is known to generate chaotic behavior.

### Jose María Muñoz-Castañeda (Universidad de Zaragoza)

#### Self adjoint extensions and heat kernel in quantum fields over bounded domains

##### Tuesday 2nd of June 2015, 10:30, UC3M, Seminar Room 2.2D08

In the last decade the experimental progress done at the nanoscale physics has allowed to build in a lab metameterials whose nanoscale physical properties do not depend on the atomic structure of the material (graphene, topological insulators, nano-ribbons, and oder bidimensional systems). From a mathematical/thoeretical point of view it is now necessary to develop rigorous formalism to study quantum physics and systems that are confined to live in compact domains with boundaries. In this talk I would like to present a review of results on quantum field theory defined over bounded domains.

### Fernando Lledó (UC3M & ICMAT)

#### Discrete magnetic Laplacians: group aspects and spectral inclusions

##### Monday 18th of May 2015, 11:00, UC3M, Seminar Room 2.2D08

In this second talk on discrete magnetic Laplacians I will recall first the definition of discrete Laplacians on oriented graphs with a magnetic vector potential field. I will also mention some of its main spectral properties as well as some interesting group theoretical aspects. Finally, I will describe a procedure to partially order discrete graphs. This is needed to apply spectral bracketing techniques for these operators.

[Joint work with Olaf Post, Univ. Trier, Germany]

### Juan Manuel Pérez Pardo (INFN, Naples)

#### Graphene and Quantum technologies: An overview

##### Tuesday 5th of May 2015, 17:00, UC3M, Seminar Room 2.2D08

Graphene is a two-dimensional layer made out of carbon atoms arranged on a honeycomb lattice. It possess outstanding properties such as mechanical stiffness, strong elasticity, or very high electrical and thermal conductance. These features, together with the fact that it is relatively easy and cheap to obtain, are responsible for the extremely fast development that graphene physics has overcome in the last 10 to 15 years.

Many of the developments are oriented to the construction and research of new materials an electronic devices (in the common usage of the word). However, from the point of view of quantum physics, graphene is a material that enhances quantum effects to the mesoscopic and macroscopic level. For instance, the Quantum Hall effect is observable at room temperature in graphene. Moreover, it presents different mesoscopic properties depending on the boundary conditions present in the probes. Low energy excitations in graphene can be modeled by massless, chiral Dirac fermions and thus graphene mimics physics of quantum electrodynamics and can serve as what is known as a Quantum Simulator.

In this talk I will concentrate on the theoretical description of graphene properties. In particular I will concentrate in the low energy limit and will point out relevant connections with the fields of quantum control and quantum computation, showing that graphene has also promising features for the physical realization of such devices.

### Jesús María Sanz Serna (UC3M)

#### Dos experimentos numéricos para pensar

##### Monday 27th of April 2015, 11:00, UC3M, Seminar Room 2.2D08

En esta charla comentaré dos experimentos numéricos que, espero, pueden llevarnos a reflexionar. El primero servirá para ilustrar las dificultades que nuestro pensamiento tiene para captar lo aleatorio. El segundo ejemplificará las posibles ventajas de los algoritmos basados en gran número de tanteos aleatorios frente a otros de más sofisticada concepción.

### Héctor Raúl Fernández Morales (UC3M)

#### Sampling Theory in Shift-Invariant Spaces: Generalizations

##### Monday 20th of April 2015, 11:00, UC3M, Seminar Room 2.2D08

In this talk we present the main results obtained in the PhD thesis of the author, entitled: Sampling Theory in Shift-Invariant Spaces: Generalizations. The classical Whittaker-Shannon-Kotel'nikov theorem states that any square integrable function with compactly supported Fourier transform is completely determined by its ordinates at a series of equally spaced points. This revolutionary result has had an enormous impact due to its applications in many scientific disciplines, but it also presents several drawbacks which leads to study sampling theory in multiple generated shift-invariant subspaces of the Hilbert space of square integrable functions. We obtain reconstruction formulas in the several dimensions case.

We also deal with the case where signals belong to a weighted Banach space. In this framework weight functions controls the decay or growth of the signals.

It is well known that the shift operator is unitary. In the previous problems we dealt with subspaces generated by this operator. A natural extension is to consider an unitary operator U on a separable Hilbert space and develop a generalized sampling theory in subspaces generated by the repeated action of U on a fixed element of the space. In order to generalize convolution systems and mainly to obtain some perturbation results, we assume that the operator U is included in a continuous group of unitary operators. We obtain some new results in this abstract setting by using techniques from frame theory, spectral theory, stationary sequences, among other branches of mathematics.

### Alberto Ibort (UC3M & ICMAT)

#### Quantum algorithms for linear systems of equations

##### Monday 23rd of March 2015, 11:00, UC3M, Seminar Room 2.2D08

In 2009 Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving a particular class of linear systems of equations that offered an exponential improvement over the best classical algorithms and, five years later, Cai et al have realised the simplest instance of it experimentally. In this talk we will discuss some of the basic ingredients and ideas behind such achievement.

### Luis Velázquez (Universidad de Zaragoza)

#### Quantum recurrence, spectral theory and Schur functions

##### Monday 16th of March 2015, 11:00, UC3M, Seminar Room 2.2D08

The exploitation of the interconnections between harmonic analysis, unitary operators and orthogonal polynomials on the unit circle goes back to works of Schur, Szegő, Geronimus, Krein, Foias, Sz.-Nagy and others. Results of much more recent vintage link these areas with quantum mechanical problems of interest in quantum information. Among them, a relation between the theory of Schur functions and a notion for recurrence in discrete time quantum systems has been recently discovered. This is the origin of a rich interplay between quantum physics and different branches of mathematics. The above connection not only provides new analytical techniques for quantum mechanical problems, but also reveals an unexpected geometrical and a topological meaning of some recurrence properties of quantum systems with surprising consequences. Even more, this relation goes both ways: The use of quantum diagrammatics has led to new relations between Schur functions and unitary operators which give the answer to unsolved problems in the theory of orthogonal polynomials.

The results that will be reviewed are the fruit of joint works with:

Jean Bourgain (IAS Princeton)

Alberto Grünbaum (UC Berkeley)

Albert Werner (Freie Universität Berlin)

Reinhard Werner, Christopher Cedzich (Leibniz Universität Hannover)

Jon Wilkening (UC Berkeley)

References:

M.J. Cantero, F.A. Grünbaum, L. Moral, L. Velázquez, Matrix valued Szegő polynomials and quantum random walks, Comm. Pure Appl. Math. 63 (2010) 464-507.

F.A. Grünbaum, L. Velázquez, A.H. Werner, R.F. Werner, Recurrence for discrete time unitary evolutions, Commun. Math. Phys. 320 (2013) 543-569.

J. Bourgain, F.A. Grünbaum, L. Velázquez, J. Wilkening, Quantum recurrence of a subspace and operator-valued Schur functions, Commun. Math. Phys. 329 (2014) 1031-1067.

C. Cedzich, F.A. Grünbaum, L. Velázquez, A.H. Werner, R.F. Werner, A quantum dynamical approach to matrix Khrushchev's formula, Comm. Pure Appl. Math. (in press), arXiv:1405.0985 [math.CA]

### Julio de Vicente (UC3M)

#### Characterizing the set of quantum correlations

##### Monday 2nd of March 2015, 11:00, UC3M, Seminar Room 2.2D08

I will try to address this talk to a general audience (of mathematicians). First, I will introduce the problem of the

characterization of the set of quantum behaviours and its relevance both foundationally (from the point of view of theoretical physics) and in applications (from the point of view of analyzing the possibilities and limitations of quantum information processing). This is a nontrivial mathematical problem for which, so far, the only systematic approach is mainly numerical. I will then present my recent results which provide simple general analytical conditions constraining this set in terms of matrix-norm inequalities. Although these conditions are necessary but not sufficient for a behaviour to be quantum, I will show their nontriviality and provide several interesting applications of them.

### Hans Munthe-Kaas (University of Bergen, Norway)

#### Computations on symmetric spaces

##### Monday 16th of February 2015, 11:00, UC3M, Seminar Room 2.2D08

Symmetric spaces (e.g. spheres and Grassman manifolds) are Riemannian manifolds with constant curvature tensor. Algebraically they are characterised as spaces with a symmetric product, inducing a Lie triple structure on the tangent space. On the group level they can be realized as quotients of Lie groups, which is closely related to Cartan decomposition of the Lie algebra; it splits in a sum of a sub algebra and a Lie triple system.

We will discuss symmetric spaces and their relationship to post-Lie algebras, and we will discuss various numerical algorithms based on Cartan decompositions and triple systems.

### David Elkouss (UCM)

#### Unbounded number of channel uses may be required to detect quantum capacity

##### Monday 2nd of February 2015, 11:00, UC3M, Seminar Room 2.2D08

The problem of transmitting data reliably over noisy communication channels is well understood for channels accurately modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities except for a handful of channels.

In this talk I will first introduce some basic ideas about quantum channels and the quantum capacity of a quantum channel and then explain some surprising properties of this quantity.

### Antonio García (UC3M)

#### Sampling-related frames in finite U-invariant subspaces

##### Monday 26th of January 2015, 11:00, UC3M, Seminar Room 2.2D08

Recently, a sampling theory for infinite dimensional U-invariant subspaces of a separable Hilbert space H where U denotes a unitary operator on H has been obtained. Thus, uniform average sampling for shift-invariant subspaces of L2(R) becomes a particular example. As in the general case it is possible to have finite dimensional U-invariant subspaces, the main aim of this talk is to derive a sampling theory for finite dimensional U-invariant subspaces of a separable Hilbert space H. Since the used samples are frame coefficients in a suitable euclidean space CN , the problem reduces to obtain dual frames with a U-invariance property.

### Alberto Ibort (UC3M & ICMAT)

#### Feynman for mathematicians

##### Monday 19th of January 2015, 11:00, UC3M, Seminar Room 2.2D08

This is a primer on Feynman’s diagrams for mathematicians. It is also a prequel of some of lasts Kurusch’s talks. More formally, this talk is about the obtention of asymptotic expansions for integrals of some ‘highly oscillatory’ functions and how combinatorics meets analysis.

Only a toy example of the theory of Feynman’s diagrams will be discussed and, time providing, some aspects of Feynman’s path integrals in Quantum Mechanics will be discussed.

### Fernando Lledó (UC3M & ICMAT)

#### On discrete magnetic Laplacians

##### Monday 12th of January 2015, 11:00, UC3M, Seminar Room 2.2D08

In the first part of the talk I will introduce discrete Laplacians on oriented graphs and mention some relations between the spectrum of the Laplacian and the topology of the underlying graph.

In the second part I will introduce a discrete analogue of a periodic magnetic Schrödinger operator (discrete magnetic Laplacian for short) and study its relation to the magnetic translation group. I will also state some questions and problems related to its spectrum.

[Joint work with Olaf Post]

### María Barbero (UC3M & ICMAT)

#### How to geometrically characterize controllability of hybrid control systems?

##### Wednesday 17th of December 2014, 09:45, UC3M, Room 2.3A04

Hybrid systems consist of an interaction between continuous dynamics and discrete events. Controls can be added both in the continuous and the discrete dynamics. Some examples of hybrid systems are given by a bouncing ball, an automobile with automatic or manual transmission, impacts, thermostat, etc. Engineers have a great interest in hybrid systems because they appear in many applications. Recently, mathematicians have focused on the geometrization of hybrid systems in order to bring more understanding to all the possible case studies. In this talk we will show the difficulties to describe controllability for hybrid systems in a similar way as for nonlinear control systems by using the tools of differential geometry. However, we will succeed in describing controllability in some fairly general situations.

References:

Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel. Hybrid dynamical systems. Princeton University Press, Princeton, NJ, 2012. Modeling, stability, and robustness.

Daniel Liberzon. Switching in systems and control. Systems & Control: Foundations & Applications. Birkhauser Boston Inc., Boston, MA, 2003.

Arjan van der Schaft and Hans Schumacher. An introduction to hybrid dynamical systems, volume 251 of Lecture Notes in Control and Information Sciences. Springer-Verlag London Ltd., London, 2000.

### Kurusch Ebrahimi-Fard (ICMAT)

#### Planar QCD from a Hopf algebra point of view

##### Wednesday 10th of December 2014, 10:00, UC3M, Room 2.3A03

We explore generating functionals for planar field theories from a Hopf algebra point of view.

### Julio de Vicente (UC3M)

#### When do certain completely positive maps take product matrices to product matrices?

##### Thursday 4th of December 2014, 10:00, UC3M, Seminar Room 2.2D08

Completely positive maps F are linear transformations on the space of square matrices of dimension d, M_d, that can be written as F(X)=\sum_k A_k^\dag X A_k for some matrices {A_k} in M_d known as Kraus operators. Motivated by some applications in quantum information and entanglement theory, we consider maps acting on M_d^{\otimes n}=M_d \otimes M_d \otimes \cdots \otimes M_d with the property that the Kraus operators are product and form a group. The goal is to characterize which product matrices can be transformed to other product matrices under this class of maps. (A product matrix X in M_d^{\otimes n} is of the form X=X_1\otimes\cdots\otimes X_n).

### Alberto López Yela (UC3M)

#### From Tomographic description of states to Clebsh-Gordan decomposition algorithm for unitary representations of groups

##### Thursday 27th of November 2014, 10:00, UC3M, Seminar Room 2.2D08

We will present an algebraic algorithm to decompose into irreducible representations unitary representations of finite groups analyzing the tomographic description of states described with that representation. Also, we will see how that algorithm can be implemented for compact Lie groups such as the SU(2) group.