Q-Math Seminar

Logo Q-Math

Danilo Jose Polo Ojito (Pontificia Universidad Catolica de Chile)

About the notion of eigenstates in C*-algebras and some application to quantum mechanics

Tuesday the 27th of June, 2023, 13:00, Room 2.2.D08

This talk is concerned with the notion of eigenstates of an operator in an abstract C*-algebra. After reviewing some basic and structural results, I will explore the possibility of reinterpreting certain typical concepts of quantum mechanics (dynamical equilibrium states, ground states, gapped states, Fermi surfaces) in terms of (algebraic) eigenstates.
Joint work with: Giuseppe De Nittis.

Link for online session (Active on request):

David Krejcirik (Czech Technical University, Prague)

Is the optimal rectangle a square?

Tuesday the 30th of May, 2023, 13:00, Room 2.2.D08

We give a light talk on an optimality of a square in geometry and physics. First, we recollect classical geometric results that the square has the largest area (respectively, the smallest perimeter) among all rectangles of a given perimeter (respectively, area). Second, we recall that the square drum has the lowest fundamental tone among all rectangular drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.

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Paolo Perrone (University of Oxford)

Categorical Probability and Information Theory

Tuesday the 9th of May, 2023, 13:00, Room 2.2.D08

Markov categories are a modern framework designed to deal with uncertainty and probability in terms of category theory. Most conceptual aspects of probability theory can be described naturally in this way, for example stochastic dependence and independence, conditioning, and conditional independence. More importantly, several theorems of probability and information theory have been recently stated, intepreted, and even proven, purely in terms of Markov categories. Among them we have the de Finetti theorem, the ergodic decomposition theorem, and a general theory of graphical models. In addition, several concepts of classical information theory, such as the notions of entropy and mutual information, and the data processing inequalities that they satisfy, can be studied using Markov categories, and can be recovered from enriched categorical principles. In this talk we give an overview of some of the concepts and results of the theory, and introduce its intrinsic graphical language.

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Video Recording

Davide Lonigro (Università di Bari)

Self-adjointness of a class of spin–boson models with ultraviolet divergences

Tuesday the 18th of April, 2023, 13:00, Room 2.2.D08

We study a class of quantum Hamiltonian operators describing a family of two-level systems (spins) coupled with a structured boson field, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences (hence, non-normalizable). Spin–spin interactions which do not modify the total number of excitations are also included. Starting from the single-atom case, and eventually reaching the general scenario, we shall provide explicit expressions for the self-adjointness domain and the resolvent operator of such models. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors by normalizable ones, for example an ultraviolet cutoff.

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Alberto Ruiz de Alarcón (Universität Tübingen)

Matrix Product Operator Algebras

Tuesday the 28th of March, 2023, 13:00, Room 2.2.D08

Understanding the entanglement structure of quantum many-body systems is one of the driving forces in theoretical physics nowadays. In particular, this has led to the use of tensor networks, the central objects of this talk. We will discuss how representations of well-known algebraic structures, e.g. weak Hopf algebras, give rise to algebras of one-dimensional tensor networks exhibiting exotic properties, known as Matrix Product Operator Algebras. We will show how to transfer results between both the algebraic and the tensor network settings and how these tools can be applied to describe Kitaev's quantum double models or classify quantum phases of matter in the open regime.

Link for online session (Active on request):

Miguel Navarro (UC3M)

Introducción a la teoría algebraica de códigos

Tuesday the 21st of February, 2023, 13:00, Room 2.2.D08

Los canales de comunicación que utilizamos en nuestro día a día son vulnerables y cada mensaje puede sufrir alteraciones durante su envío. Por ello, tener garantías de que un mensaje recibido es correcto o, en caso contrario, poder detectar o corregir los errores producidos es de gran importancia en cualquier proceso de comunicación.

La teoría algebraica de códigos surge ante esta necesidad a mediados del siglo pasado y, desde entonces, se desarrolla como una teoría matemática relacionada con el álgebra, la geometría y la combinatoria. Al utilizar códigos detectores y correctores de errores, añadimos información redundante a cada mensaje. De esta forma, incrementamos las diferencias entre mensajes distintos y, en consecuencia, detectar (e incluso corregir) cierta cantidad de errores se vuelve más fácil.

En esta charla daremos una introducción a la teoría algebraica de códigos, haciendo especial énfasis en algunas familias de códigos de distancia máxima (aquellos con mayor capacidad correctora de errores). Empezaremos con los códigos bloque clásicos y estudiaremos su evolución hasta llegar a los códigos de subespacio y los códigos flag, en los que se centra mi investigación.

Link for online session (Active on request):

Laiachi EL Kaoutit, Universidad de Granada

Grupoides, algebroides de Hopf y algebroides de Lie

Thursday the 9th of February, 2023, 13:00, Room 2.2.D08

Grupoides, algebroides de Hopf y algebroides de Lie son objetos matemáticos que aparecen en diferentes areas de matemática y física teorica. Al igual que los grupos, algebras de Hopf y álgebras de Lie, estos objetos son dignos de investigar y sobre todo lograr entender como se relacionan entre si.

En la primera parte de la charla vamos a recordar los funtores que conectan los grupos con las álgebras de Lie, pasando por las álgebras de Hopf de las funciones representativas (es decir, recordar el caso clásico: el de un sólo objeto). En la segunda parte, introducimos las nociones necesarias sobre dichos objetos junto con algunos ejemplos básicos. Luego pasaremos a dar ciertas indicaciones de como construir funtores entre los grupoides y los algebroides de Lie, usando esta vez los algebroides de Hopf de funciones representativas. Por último, vamos a intentar exponer algunas aplicaciones, así como ciertos problemas abiertos.

Link for online session (Active on request):

Florio M. Ciaglia (UC3M)

Can Cencov meet Petz?

Friday the 27th of January, 2023, 13:00, Room 2.2.D08

Cencov's theorem on the uniqueness of the Fisher-Rao metric tensor and Petz's classification of monotone metric tensors are considered the cornerstones of classical and quantum information geometry, respectively. I will review and discuss these beautiful results with particular attention to a categorical formalization of Petz's classification in analogy with Cencov's ideology, and I will then argue about a possible unification in the framework of Jordan algebras.

Link for online session (Active on request):

Satoya Imai (University of Siegen)

Geometry of quantum entangled states

Friday the 25th of November, 2022, 13:00, Room 2.2.D08

In this talk, I will first give an introduction of quantum entanglement in the language of quantum information. Then I will explain the problem of whether a given state is entangled or not. Finally I will provide several methods to detect entanglement from geometrical viewpoints [1-4].

[1] S Imai, N Wyderka, A Ketterer, O Gühne, Physical Review Letters 126 (15), 150501
[2] XD Yu, S Imai, O Gühne, Physical Review Letters 127 (6), 060504
[3] A Ketterer, S Imai, N Wyderka, O Gühne, Physical Review A 106 (1), L010402
[4] some ongoing projects.

Link for online session:

Alejandro Pozas-Kerstjens (ICMAT)

Quantum-inspired solutions for privacy leaks in machine learning

Friday the 11th of November, 2022, 13:00, Room 2.2.D08

Vast amounts of data are routinely processed in machine learning pipelines, every time covering more aspects of our interactions with the world. However, the quest for performance is leaving other important aspects, such as privacy, on the side. For example, when the models processing the data are made public, is the safety of the data used for training it guaranteed? This is a question of utmost importance especially when processing sensitive data such as medical records.
In this talk, I will argue and practically illustrate that insights in quantum information, concretely coming from the tensor network representations of quantum many-body states, can help in devising better privacy-preserving machine learning algorithms. In the first part, I will show that standard neural networks are vulnerable to a type of privacy leak that involves global properties of the data used for training, thus being a priori resistant to standard protection mechanisms. In the second, I will show that tensor networks, when used as machine learning architectures, are invulnerable to this vulnerability. The proof of the resilience is based on the existence of canonical forms for tensor networks. Given the growing expertise in training tensor networks and the recent interest in tensor-based reformulations of popular machine learning architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed when using machine learning on sensitive data.

Nicolai Rothe (Bergische Universität Wuppertal)

De Sitter Solutions to the Semiclassical Einstein Equation in Cosmology

Friday the 21st of October, 2022, 13:00, Room 2.2.D08

We present a characterization of cosmological de Sitter solutions to the semiclassical Einstein equation (SCE) with a free scalar quantum field in the Bunch-Davies state. In this setting, the SCE may be viewed as a (non-dynamic) consistency equation for the parameters of the model. Our approach allows to identify parameter settings in which there exist multiple cosmological de Sitter solutions with expansion rates $H$ differing by several (in fact, arbitrarily many) magnitudes for only one field theory. By these observations a quantum field is, in principle, capable to drive both an inflationary phase (approximated by the large-$H$ solution) and a dark energy-dominated phase (approximated by the small-$H$ solution) in the expansion of the universe.

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Julio de Vicente (UC3M)

Genuine multipartite entanglement of quantum states in the multiple-copy scenario

Friday the 14th of October, 2022, 13:00, Room 2.2.D08

Genuine multipartite entanglement (GME) is considered a powerful form of entanglement since it corresponds to those states that are not biseparable, i.e. a mixture of partially separable states across different bipartitions of the parties. In this talk I will study this phenomenon in the multiple-copy regime, where many perfect copies of a given state can be produced and controlled. In this scenario the above definition leads to subtle intricacies as biseparable states can be GME-activatable, i.e. several copies of a biseparable state can display GME. I will show, however, that the set of GME-activatable states admits a simple characterization: a state is GME-activatable if and only if it is not partially separable across one bipartition of the parties. This leads to the second question of whether there is a general upper bound in the number of copies that needs to be considered in order to observe the activation of GME, which will be answered in the negative. In particular, by providing an explicit construction, it can be proven that for any number of parties and any number $k\in\mathbb{N}$ there exist GME-activatable multipartite states of fixed (i.e. independent of $k$) local dimensions such that $k$ copies of them remain biseparable. This is joint work with Carlos Palazuelos.

Zhen-Peng Xu (Universität Siegen)

Quantum networks cannot generate graph states with high fidelity

Friday the 23rd of September, 2022, 13:00, Room 2.2.D08

Quantum networks lead to novel notions of locality and correlations which are not well understood. An important problem concerns the question which states can be experimentally prepared with a given network structure and which not. By exploiting the inflation technique and symmetry analysis, we prove that all multi-qubit graph states arising from a connected graph cannot originate from any bipartite network. Moreover, the fidelity of a multi-qubit graph state and any network state cannot exceed $9/10$. Similar results can also be established for a large class of multi-qudit graph states. More specifically, the fidelity of any dimensional graph state with constant multiplicities and states prepared in a bipartite network cannot exceed $0.95495$.