## Q-Math Seminar

### Rob Corless (University of Western Ontario, Canada)

#### Gamma and Factorial in the Monthly

##### Wednesday the 21st of June, 2017, 11:00, UC3M, Seminar Room 2.2D08

Since its inception in the 19th century, the American Mathematical Monthly has published over fifty papers on the Gamma function or equivalently the factorial function. Over half of these were on Stirling’s formula. We survey these papers, which include a Chauvenet prizewinning paper by Philip J. Davis and a paper by the Fields medallist Manjul Bhargava, and highlight some features in common. We also identify some surprising gaps and attempt to fill them, especially on the “inverse Gamma function”.

This is joint work with the late Jonathan M. Borwein.

### Mahul Pandey (Indian Institute of Science, Bangalore)

#### Quantum Phases in an SU(2) YM Matrix Model Coupled with Fermions

##### Monday the 12th of June, 2017, 11:00, UC3M, Seminar Room 2.2D08

By investigating the $SU(2)$ Yang-Mills matrix model coupled to fundamental fermions in the adiabatic limit, we demonstrate quantum critical behaviour at special corners of the gauge field configuration space. The quantum scalar potential for the gauge field induced by the fermions diverges at the corners, and is intimately related to points of enhanced degeneracy of the fermionic Hamiltonian. This in turn leads to superselection sectors in the Hilbert space of the gauge field, the ground states in different sectors being orthogonal to each other. These we interpret as different quantum phases in the theory.

### Fernando Lledó (UC3M & ICMAT)

#### The theory of superselection sectors and properly infinite algebras

##### Tuesday the 6th of June, 2017, 11:00, UC3M, Seminar Room 2.2D08

In quantum physics a superselection rule can be understood as any restriction on the set of observables of a theory. In this talk I will sketch some aspects of this theory including the Doplicher-Haag-Roberts (DHR) selection principle and the construction of the field algebra out of the set of observables. I will stress the importance of properly infinite algebras in this construction.

### Alberto Ibort (UC3M & ICMAT)

#### A categorical description of Schwinger’s algebra of selective measurements II: 2-groupoid structure

##### Wednesday the 24th of May, 2017, 12:00, UC3M, Seminar Room 2.2D08

In these two talks we will try to explain part of our recent work on the algebraic structure of Schwinger’s algebra of selective measurements. We will show that a categorical language suits the description of the structure of Schwinger’s algebra and its representations. It will be conjectured that the structure of Schwinger’s algebra is that of a 2-groupoid, offering in this way a new approach to the mathematical foundations of Quantum Mechanics.

The first talk will be devoted to offer a succinct description of the elements of category theory needed in our description of the algebraic structure of Schwinger’s algebra. The second talk will be devoted to review the theory of Schwinger’s selective measurements and their algebraic properties.

### Alberto Ibort (UC3M & ICMAT)

#### A categorical description of Schwinger’s algebra of selective measurements I: A primer on category theory

##### Monday the 8th of May, 2017, 11:00, UC3M, Seminar Room 2.2D08

In these two talks we will try to explain part of our recent work on the algebraic structure of Schwinger’s algebra of selective measurements. We will show that a categorical language suits the description of the structure of Schwinger’s algebra and its representations. It will be conjectured that the structure of Schwinger’s algebra is that of a 2-groupoid, offering in this way a new approach to the mathematical foundations of Quantum Mechanics.

The first talk will be devoted to offer a succinct description of the elements of category theory needed in our description of the algebraic structure of Schwinger’s algebra. The second talk will be devoted to review the theory of Schwinger’s selective measurements and their algebraic properties.

### Antonio García (UC3M)

#### Sampling formulas involving differences in shift-invariant subspaces: a unified approach

##### Monday the 24th of April, 2017, 11:00, UC3M, Seminar Room 2.2D08

Successive differences on a sequence of data help to discover some smoothness features of this data. This is the reason for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper is to mimic them to a sequence of regular samples of a function in a shift-invariant subspace allowing its stable recovery. A suitable expression for the functions in the shift-invariant subspace by means of an isomorphism with the L^2(0,1) space is the key to identify the simple pattern followed by the dual Riesz bases involved in the derived formulas. The paper contains examples illustrating different non-exhaustive situations including also the two-dimensional case.

### John Stewart Fabila Carrasco (UC3M)

#### Lagunas espectrales y el Laplaciano magnético en grafos

##### Monday the 27th of March, 2017, 11:00, UC3M, Seminar Room 2.2D08

El Laplaciano es uno de los objetos más estudiados en Teoría de Grafos. Una de las características más interesantes de este operador es su espectro. Cuando el grafo es finito, el espectro del Laplaciano esta formado por un número finito de valores propios, pero cuando el grafo es infinito, el espectro es más complejo. Una generalización de este operador es el Laplaciano magnético.

En esta charla estudiaremos las lagunas espectrales del Laplaciano de grafos infinitos periódicos. Estos grafos tienen la propiedad de "cubrir" grafos finitos. Encontraremos una relación entre estudiar el Laplaciano en el grafo infinito y estudiar el Laplaciano magnético en el grafo finito. Esta relación nos la dará la Teoría de Floquet.

### Giancarlo Garnero (INFN & Università di Bari)

#### A quantum particle in a cavity with alternating boundary conditions

##### Monday the 13th of March, 2017, 11:00, UC3M, Seminar Room 2.2D08

We consider the quantum dynamics of a non-relativistic free particle moving in a cavity and we analyze the effect of a rapid switching between two different boundary conditions. We show that this procedure induces, in the limit of infinitely frequent switchings, a new effective dynamics in the cavity related to a novel boundary condition. We explicitly compute the novel boundary condition in terms of the two initial ones. With this procedure we define a dynamical composition law for boundary conditions.

### Giuseppe Marmo (INFN Naples)

#### Dynamical vector fields on the manifold of quantum states III

##### Monday the 27th of February, 2017, 11:00, UC3M, Seminar Room 2.2D08

For finite systems, quantum states are a subset of the manifold of Hermirtian operators. The positivity condition on states requires the subset of states to be a manifold with corners. The normalization condition restricts the positive operators to the affine subspace of unit trace operators. The presence of corners requires suitable care to deal with Poisson tensors and Jordan tensors. We shall introduce Hamiltonian gradient and Kraus vector fields to deal with Markovian and non-Markovian evolution.

### Giuseppe Marmo (INFN Naples)

#### Dynamical vector fields on the manifold of quantum states II

##### Monday the 20th of February, 2017, 11:00, UC3M, Seminar Room 2.2D08

For finite systems, quantum states are a subset of the manifold of Hermirtian operators. The positivity condition on states requires the subset of states to be a manifold with corners. The normalization condition restricts the positive operators to the affine subspace of unit trace operators. The presence of corners requires suitable care to deal with Poisson tensors and Jordan tensors. We shall introduce Hamiltonian gradient and Kraus vector fields to deal with Markovian and non-Markovian evolution.

### Giuseppe Marmo (INFN Naples)

#### Dynamical vector fields on the manifold of quantum states I

##### Monday the 13th of February, 2017, 11:00, UC3M, Seminar Room 2.2D08

For finite systems, quantum states are a subset of the manifold of Hermirtian operators. The positivity condition on states requires the subset of states to be a manifold with corners. The normalization condition restricts the positive operators to the affine subspace of unit trace operators. The presence of corners requires suitable care to deal with Poisson tensors and Jordan tensors. We shall introduce Hamiltonian gradient and Kraus vector fields to deal with Markovian and non-Markovian evolution.

### Carlos González-Guillén (UPM)

#### Spectral gap of random quantum channels

##### Friday the 27th of January, 2017, 12:00, UC3M, Seminar Room 2.2D08

Quantum channels (completely positive linear maps) are the most general operation that can be applied to quantum systems. Thus, their study is of main interest in many areas of the quantum sciences. In particular, they appear naturally in the study of entanglement, quantum correlations, communication tasks, etc.

In this talk, we will concentrate in the study of random quantum channels. In particular, we will discuss how to sample quantum channels with respect to a pushforward measure of the Haar measure over the unitary group. For this particular way of sampling quantum channels, we will show that they have generically a separation between the first and second singular values. We will see that, in a broad sense, this result can be seen as a generalisation of Hasting's result for a set of random unital channels (arXiv:0706.0556). Moreover, we will discuss different applications of this result to study entropy and correlations in 1D tensor networks. Work in progress.

### Florio M. Ciaglia & Fabio Di Cosmo (INFN & Università degli Studi di Napoli)

#### Metric tensors on the space of quantum states

##### Friday the 16th of December, 2016, 12:00, UC3M, Seminar Room 2.2D08

According to methods well-known in Information Geometry, it is possible to define tensors starting from two-point functions called Divergences which generalize the concept of distances to the asymmetric case. Recently, we have proposed to define a canonical divergence as the solution of the Hamilton-Jacobi equation associated to a suitably defined dynamical system.

By moving these considerations to the quantum setting, in this talk we will present a way to define metric tensors on the space of states of a finite-dimensional quantum system. First of all, we will present the Geometry of the space of Quantum States, in order to make clear similarities and differences among classical and quantum settings. In particular, the latter is a stratified manifold, each strata being the orbit of a GL(n,$\mathbb{C}$) non-linear action. Therefore, in order to avoid difficulties arising from the differential structure of this space, we will define a divergence function and a metric tensor in a bigger space which is the tangent bundle $TGL(n,\mathbb{C})$ over the Lie group $GL(n,\mathbb{C})$: in a certain sense it will play the role of a "universal" space, common to each strata. By defining a suitable dynamical system on this space we will obtain a canonical divergence as solution of the related Hamilton-Jacobi equation. In particular we will show that this function is the pull-back of a function on each stratum and the deriving metric tensor is the pull-back of a symmetric tensor on the base space.

In order to make contact with some explicit expression we will present the simplest example, which is a two-level quantum system, that is a q-bit. In particular the metric obtained in this case is conformally related to one of the Petz metric tensors, which are monotone with respect to stochastic maps.

In the end we will present the form of the tensor in the general case and we will show that there exists a limiting procedure according to which this tensor can be extended to the other strata without developing singularities.

### Alberto Ibort (UC3M & ICMAT)

#### The hidden subgroup problem and sampling theory

##### Friday the 25th of November, 2016, 12:00, UC3M, Seminar Room 2.2D08

We will examine the hidden subgroup problem, that lies behind some of the most important quantum algorithms, and its relation with the sampling theory applied to quantum systems recently developed by A. García, AI and M.A. Hernández-Medina.

### Julio de Vicente (UC3M)

#### Completely positive maps and entanglement manipulation

##### Friday the 11th of November, 2016, 12:00, UC3M, Seminar Room 2.2D08

The paradigm of local operations and classical communication (LOCC) plays a key role in entanglement theory in quantum information. First, these provide the most general possible protocols to manipulate this resource in practical scenarios. Second, and most importantly, these constitute the free operations in the resource theory of entanglement. Thus, they induce an operationally meaningful ordering in the set of entangled states and provide the means to construct entanglement measures. Mathematically speaking, LOCC maps are a subset of completely positive maps (on matrices), the latter providing the most general form of quantum dynamics. However, LOCC is mathematically subtle and it is in general hard to characterize when such transformations are possible. In this seminar I will put the stress on presenting basic definitions and useful techniques in this context rather than on presenting my own results. In particular, I will highlight the differences between relevant classes of completely positive maps in this setting: separable operations, finite-round LOCC and infinite-round LOCC.

### Juan Manuel Pérez Pardo (UC3M)

#### Quantum mechanics and information geometry

##### Friday the 28th of October, 2016, 12:00, UC3M, Seminar Room 2.2D08

It was realised by S. Amari and cowokers, cf. [1] and references therein, that differential geometry is a useful means to study statistical models. In particular, they possess the structure of a Riemannian manifold, with metric the Fisher-Rao metric, and a pair of so-called dual connections. The geometric formulation of Quantum Mechanics allows to use those ideas and, therefore, to study its statistical structure from a geometric viewpoint. It turns out that the Fisher-Rao metric is ``built-in'' in a natural way within Quantum Mechanics. We will exploit these geometric relationship and study the role that well-developed notions such as complete integrability and totally geodesic embeddings play in this context.

[1] S. Amari and H. Nagaoka, Methods of Information Geometry, Oxford University Press (2000).