## Q-Math Seminar

### Diego Martínez (Universidad Autónoma de Madrid)

#### Descomposiciones paradójicas en operadores y espacios de Hilbert

##### Monday 13th of June 2016, 12:00, UC3M, Seminar Room 2.2D08

En teoría de grupos es bien conocido que la existencia de descomposiciones paradójicas, la no existencia de medidas invariantes o la no existencia de aproximaciones tipo Følner son equivalentes. Esta equivalencia se puede probar en otros campos, como espacios métricos discretos o álgebras sobre cuerpos conmutativos K. En teoría de operadores, sin embargo, sólo han aparecido las aproximaciones tipo Følner y la existencia de trazas invariantes. En esta charla analizaremos el concepto de descomposición paradójica de la acción de un operador lineal y acotado en un espacio de Hilbert, tratando de encontrar su relación con los conceptos ya asentados en la teoría.

### Angelo Lucia (Universidad Complutense de Madrid)

#### Stability and area law for rapidly mixing dissipative quantum systems

##### Monday 16th of May 2016, 12:00, UC3M, Seminar Room 2.2D08

In condensed matter and quantum information we usually describe many-body systems as tensor product of finite-dimensional Hilbert spaces over a graph or lattice structure. Measurable quantities are encoded into a state, a positive and trace one operator. Operations that preserve states are completely positive and trace preserving maps, also called quantum channels. For a continuous time description, we consider dynamical semigroups of quantum channels, that can be seen as a non-communtative generalisation of stochastic processes. They model open evolutions where the systems interacts with an environment.

We will consider bounds on the mixing time of this processes which scale logarithmically with the system sizes. Mixing time is the time needed for any input state to converge to a small neighborhood of the fixed point. With this assumption we are able to prove a number of interesting properties of the evolution: that it is stable under local perturbations, and that its fixed point satisfy an area law for the mutual information.

### Piergiulio Tempesta (Universidad Complutense de Madrid & ICMAT)

#### Group Entropies, Complex Systems and Number Theory

##### Monday 9th of May 2016, 12:00, UC3M, Seminar Room 2.2D08

We will show that an intrinsic group-theoretical structure is at the heart of the notion of entropy. This structure emerges when imposing the requirement of composability of an entropy with respect to the union of two statistically independent systems. A new formulation of the celebrated Shannon-Khinchin set of axioms is proposed, obtained by replacing the additivity axiom with that of composability.

The theory of formal groups offers a natural language for our group-theoretical approach to generalized entropies. In this settings, the known entropies can be encoded into a general trace-form class, the universal-group entropy (so called due to its relation with the Lazard universal formal group of algebraic topology).

We shall also prove that Renyi's entropy is the first example of a new family of non trace-form entropies, of potential interest in the theory of complex systems, called the Z-entropies. Each of them is composable and, in particular, generalizes simultaneously the entropies of Boltzmann and Renyi (obtained under suitable limits). The information theoretical content of composable entropies is shown to be a byproduct of their underlying group structure.

The theory of group entropies will also be related with L-series of analytic number theory and with generalized Bernoulli polynomials.

### Julio Moro (UC3M)

#### Eigenvalue perturbation expansions and structured condition numbers

##### Monday 25th of April 2016, 12:00, UC3M, Seminar Room 2.2D08

The aim of this talk is to present the basics of classic first order perturbation theory for eigenvalues of additively perturbed matrices: given a possibly multiple eigenvalue of an unperturbed matrix A, we identify the leading term in the asymptotic expansions in \epsilon of eigenvalues of A + \epsilon B, where \epsilon is a small, real parameter. Such expansions provide local information on the perturbation behavior of the eigenvalue, which is usually translated into the spectral condition number, a local measure of the sensitivity of that particular eigenvalue to small changes in the matrix.

When the matrix A belongs to some relevant class of structured matrices, one may need to restrict perturbations to only those preserving that specific structure. This gives rise to the so-called structured condition number, which measures the sensitivity of the eigenvalue to only structure-preserving perturbations. If time allows, we will define structured condition numbers, and compare them to unstructured ones for several classes of matrices defined through symmetries with respect to indefinite scalar products.

### Julio de Vicente (UC3M)

#### Completely positive maps and quantum resource theories: Schur maps and quantum coherence

##### Monday 4th of April 2016, 12:00, UC3M, Seminar Room 2.2D08

In the last decades it has been realized that the particularities of quantum theory can be exploited to construct revolutionary technologies. Hence, certain quantum features such as entanglement or nonlocality are considered as a resource to implement informational or computational tasks. This has led to the development of quantum resource theories in order to build a rigorous framework to understand the ultimate possibilities and limitations of quantum information processing. Given a particular type of quantum resource, these theories aim at characterizing all quantum states with this property, at identifying all possible protocols for its manipulation and at providing measures for its quantification. The most general transformations possible in practice among quantum states are given by completely positive (CP) maps. Thus, mathematically, the analysis of quantum resource theories boils down to the study of the particular type of CP maps the corresponding physical setting allows to implement. In the last years the superposition principle (or coherence) has been identified as a potential quantum resource which is believed to play a nontrivial role in the outstanding efficiency of certain biological processes such as photosynthesis. The theoretical ground for the construction of a resource theory of quantum coherence is currently being discussed by different researchers.

In this talk I will present a gentle introduction to CP maps on matrix algebras and quantum resource theories. I will then focus to the particular case of coherence and introduce some of my recent results obtained in collaboration with Alexander Streltsov (Freie Universitat Berlin). We show that the class of CP maps relevant in this context are Schur maps (the name comes from the fact that the action of this maps can be succinctly written in terms of the Schur or Hadamard product). We then use this characterization to obtain several different results on the possibilities and limitations of such resource theory of coherence for state manipulation.

### Alberto Ibort (UC3M & ICMAT)

#### A tale of two entropies and a ghost

##### Monday 7th of March 2016, 12:00, UC3M, Seminar Room 2.2D08

A gentle discussion on some fundamental aspects around Gibbs-Boltzmann and Shannon’s entropy notions will be offered. The ever present Second Law of Thermodynamics will be surveyed from a mathematical perspective and its far reaching consequences in the presence of extreme gravitational events will be assessed.

### Christian Mehl (Technische Universität Berlin)

#### Perturbation theory for Hermitian pencils

##### Monday 22nd of February 2016, 12:00, UC3M, Seminar Room 2.2D08

Symmetric or Hermitian pencils can be considered as a generalization of several classes of matrices with symmetry structures, like for example symmetric or Hermitian matrices, or Hamiltonian matrices.

In this talk, we consider the problem of finding the smallest Hermitian perturbation so that a given value is an eigenvalue of the perturbed Hermitian pencil. The norm of this perturbation is then called the structured eigenvalue backward error of the given value. The answer is well known for the case that the eigenvalue is real, but in the case of nonreal eigenvalues, the problem is more involved.

We give a complete answer to the question by reducing the problem to an eigenvalue minimization problem of Hermitian matrices depending on two real parameters. We will see that the structured eigenvalue backward error of complex nonreal eigenvalues may be significantly different from the corresponding unstructured eigenvalue backward error - which is in contrast to the case of real eigenvalues, where the structured and unstructured backward errors coincide. This effect is partly caused by the so-called sign characteristic, an additional structural invariant besides eigenvalues, eigenvectors, and invariant subspaces.

### Miguel Ángel Hernández Medina (Universidad Politécnica de Madrid)

#### Sampling associated with a unitary representation of a finite group

##### Monday 8th of February 2016, 12:00, UC3M, Seminar Room 2.2D08

The aim of the present talk is to present the first steps into the construction of a rigorous and broad enough sampling theory for quantum systems inspired directly in the classical sampling theorem by Nyquist-Shannon. An extension of a generalized sampling formula based in the notion of U-invariant subspaces will be considered. The notion of $U$-invariant subspaces corresponds to the choice of the Abelian group of integers Z and the unitary representation provided by $U(n) = U^n$ where $U$ is a given unitary operator defined on a Hilbert space H. In this talk, we will consider a finite group G (not neccesarily conmmutative) and a unitary representation U: G -> U(H). Thus, the theory presented can be considered as a non-commutative extension of the standard classical sampling theory.

### Fernando Lledó (UC3M & ICMAT)

#### Amenability, paradoxical decompositions and applications to physics

##### Monday 18th of January 2016, 12:00, UC3M, Seminar Room 2.2D08

There is a classical mathematical theorem (based on the work by Banach and Tarski) that implies the following shocking statement: “An orange can be divided into finitely many pieces, these pieces can be moved and rearranged in such a way to yield two oranges of the same size as the original one.” In 1929 J. von Neumann recognized that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability (“promediabilidad” in Spanish or “Mittelbarkeit” in German). Since then, the notion of amenability has become ubiquitous in mathematics.

In this talk I will introduce first the notion of amenability and paradoxical decomposition in very different mathematical situations: in the context of groups, discrete metric spaces and, time permiting, K-algebras. In the second part of the talk I will speculate about the possibility of using amenability arguments in physical situations.

### Antonio García (UC3M)

#### Modeling sampling in tensor products of unitary invariant subspaces

##### Monday 14th of December 2015, 12:00, UC3M, Seminar Room 2.2D08

The use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of L2(R) and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.

### Alberto López Yela (UC3M)

#### On the tomographic description of quantum systems: theory and applications

##### Monday 23rd of November 2015, 12:00, UC3M, Seminar Room 2.2D08

This talk will be a trial of the thesis defense corresponding to the title above. It will be made an introduction of the CAT process for reconstructing images of sections of scanned bodies and will be shown a natural transition of this technique to reconstruct the quantum state of a radiation source by means of the Homodyne Detection in Quantum Optics.

Using the Homodyne Detection as motivation, we will present a tomographic description of Quantum Mechanics on C*-algebras thanks to the GNS construction, and we will show that the theory can be split in two parts: a Generalized Sampling Theory and a Generalized Positive Transform.

Once we get a clear theory, we will show a particular instance of it by means of the symmetry group that underlies the quantum sysyem parametrized by a Lie group. Also, we will present a special case of quantum states related with groups that we will call adapted states to a subgroup, that will be the main tool of an algorithm that we will explain later to decompose any finte dimensional unitary representation in irreducible representations.

To finish the presentation, we will present a tomographic description in Quantum Field Theory based on Wightman-Streater axioms.

### Joan Claramunt (UAB & ICMAT)

#### Paradoxical decompositions in groups and algebras

##### Monday 9th of November 2015, 12:00, UC3M, Seminar Room 2.2D08

Almost every mathematician in these times has heard about the Banach-Tarski paradox, which roughly states that you can cut (in some bizarre way) an orange into finitely many pieces and rearrange them in order to get two oranges of exactly the same size as the original one! The mathematical formulation of this fact is that we can decompose the unit ball $B^1 \subset \mathbb{R}^3$ into finitely many pieces and then let the group of rotations $SO(3)$ of $\mathbb{R}^3$ act on these pieces in such a way that some isometries (and translations if we want) give the desired rearrangement of the pieces in order to get $2$ unit balls: $B^1 \stackrel{SO(3)}{\longrightarrow} B^1 \sqcup B^1$.

This can be formulated in a general setting: for a given set $X$ where a finitely generated discrete group $G$ acts, we can talk about paradoxical decompositions of this set to mean that we can break such a set $X$ in finitely many disjoint pieces $A_1 \sqcup \cdots \sqcup A_n \sqcup B_1 \sqcup \cdots \sqcup B_m$ in such a way that we can recover the whole space with some translates of such a partition: $g_1A_1 \sqcup \cdots \sqcup g_nA_n$ and $h_1B_1 \sqcup \cdots \sqcup h_mB_m$. We say $X$ is $G$-paradoxical if such a decomposition exists, and we also say $G$ itself is paradoxical if it admits such a decomposition by considering the left-action on itself.

In the first part of the talk, we will comment on the well-known Tarski's Theorem, which characterizes when a set $X$ is not $G$-paradoxical by means of the existence of an invariant probability measure $\mu \colon \wp(X) \rightarrow [0,1]$. This theorem consitutes the beginning of the definition of an amenable group: we say $G$ is amenable if it is not paradoxical or, equivalently, that there exists such an invariant probability measure $\mu \colon \wp(G) \rightarrow [0,1]$.

Some recent work has been done when trying to transfer the definition of paradoxical decompositions into more general mathematical objects, such as metric spaces and algebras. We will devote the last part of this talk by transfering such a definition into the world of algebras, so we are going to talk about paradoxical decompositions in algebras, and present the equivalent of Tarski's Theorem, characterizing amenable algebras in terms of such decompositions.

### Kurusch Ebrahimi-Fard (ICMAT)

#### On the combinatorics of planar Green's functions (II)

##### Monday 2nd of November 2015, 12:00, UC3M, Seminar Room 2.2D08

The aim of this series of two talks is to outline the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green’s functions. The key object is the natural Hopf algebra of non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green’s functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between connected and one-particle irreducible Green’s functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to sources can be translated into the language of growth operations on rooted trees.

### Kurusch Ebrahimi-Fard (ICMAT)

#### On the combinatorics of planar Green's functions (I)

##### Monday 19th of October 2015, 12:00, UC3M, Seminar Room 2.2D08

The aim of this series of two talks is to outline the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green’s functions. The key object is the natural Hopf algebra of non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green’s functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between connected and one-particle irreducible Green’s functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to sources can be translated into the language of growth operations on rooted trees.