Here - Cimpa

CIMPA-MOROCCO research school on
Operator Theory and The Principles of Quantum Mechanics
8-17 September, 2014
Abstracts of Courses, Lectures and Talks
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Abstracts of the mini-courses and invited lectures
Abdelmalek Azizi
University Mohammed Premier, Oujda, Morocco
Public Lecture: Les idées modernes de la cryptographie
numérique dans les pages d’histoire du Maroc
Depuis des temps très reculés, l’Homme avait utilisé diverses méthodes et
techniques pour cacher un message, comme les méthodes de cryptographie
ou les méthodes de stéganographie. Plusieurs civilisations avaient inventé
leurs propres méthodes de communication secrètes, en particulier les Marocains avaient développé des méthodes de cryptographie et de stéganographie.
Avant le 16ème siècle, ils avaient utilisé la notion de décomposition d’un entier naturel en produit de deux entiers, les fonctions de Hachages, la signature numérique et avaient introduit la cryptographie arithmétique. Ce qui
est encore utilisé dans la cryptographie moderne comme dans la méthode
RSA. Dans cette conférence je donnerais un survol de l’histoire de la cryptographie au Maroc, un aperçu sur certaines méthodes de la cryptographie moderne (RSA, ) et une comparaison entre les outils utilisés par ces
méthodes. Je finirais par parler de leurs faiblesses devant la cryptographie
quantique.
Mohamed Bennai
University Hassan II-Mohammedia, Casablanca, Morocco
Lecture: L’implémentation des portes logiques en
électrodynamique en Cavité
Récemment, le développement de nouveaux systèmes de traitement et de
stockage de l’information a conduit à l’introduction des concepts de la physique
quantique dans l’élaboration des algorithmes et des systèmes de traitement
d’information. Nous nous intéressons aux très récents développements de
cette nouvelle thématique, initié par le physicien R. Feynman, où l’information
peut être codée dans les états quantiques des photons, des atomes, ou des
ions. Le traitement quantique de l’information, ainsi développé va montrer une très grande puissance et une forte capacité à résoudre certains
problèmes d’intérêt pratique, tels que ceux de la cryptographie. Cette
découverte majeure a ouvert la voie vers une révolution dans le domaine
de la cryptographie en montrant par exemple que le fameux code RSA
peut être brisé. Cette nouvelle théorie de l’information a introduit de nouvelles méthodes de calcul et procédures offrant une grande possibilité pour
élaborer des protocoles cryptographiques présentant une grande sécurité
de transmission de l’information, exploitant le concept de l’intrication ou
l’entanglement. Nous nous sommes intéressés à l’étude de la nouvelle théorie
d’électrodynamique quantique en cavité, en montrant comment certaines
portes logiques peuvent être implémentées sur des systèmes réels de qubits,
tels que l’implémentation dans les circuits quantiques supraconducteurs,
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via l’électrodynamique quantique en Cavité (QED). Dans ce contexte, nous
avons introduit un système de deux atomes en interactions en Cavité QED
et nous avons calculé explicitement les différentes quantités réalisant cette
l’implémentation. Nous proposons en particulier l’implémentation de la
porte de phase d’un seul qubit simultanément contrôlant n qubits en circuit
QED. Dans notre schéma, nous utilisons un système dans lequel les qubits
superconducteurs sont couplés à un résonateur. L’interaction qubit-qubit a
été considérée.
Ingemar Bengtsson
University of Stockholm, Sweden
Mini-course: A geometric approach to quantum entanglement
The geometry of a set is determined by the transformations we allow. In
classical probability theory they are stochastic maps, in quantum theory
they are completely positive maps. I will explain these connections, and
the role played by entanglement in determining the geometry of the set of
quantum states. Afterwards we will study the geometry of the simplest
case — two entangled qubits — in some detail, and then we end with the
connection between maximally entangled states and unitary operator bases.
Dagmar Bruss
University of Düsseldorf, Germany
Mini-course: Introduction to quantum information
This course will start with basic notions of quantum information, namely
qubits, composite sytems, no-cloning theorem, measurements, and some information theoretic concepts. Important quantum information processing
protocols such as quantum teleportation, superdense coding and quantum
cryptography will be explained. Two models for quantum computation will
be covered: quantum gates and the circuit model on one side, and oneway quantum computing on the other side. The effect of quantum noise
(described via quantum channels) will finally be discussed.
Adán Cabello
University of Sevilla, Spain
Mini-course: The principles of Quantum Mechanics
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The course will have two parts.
The first (1. The rules of Quantum Mechanics: The surprising connection
between nature and Hilbert spaces), will be a review of the rules of Quantum
Mechanics, with emphasis in the one-to-one correspondence between physical procedures and mathematical objects (e.g., between repeatable tests and
self-adjoint operators), and the absence of ”conceptual problems” when the
”quantum state of a system” is accepted as a state of the observers’ knowledge about future experiments on the system.
The second part (2. Quantum contextuality, nonlocality and the quest for
the physical principles of Quantum Mechanics), will cover some recent attempts to find physical symmetries and physical principles explaining the
rules of Quantum Mechanics, with emphasis in those which explain why
Quantum Mechanics is as contextual and non-local as it is.
Alexei Grinbaum
Laboratoire LARSIM, CEA-Saclay
Mini-course: Quantum logical structures
We describe the history of applying mathematical constructions as descriptions of the fundamental structure of quantum mechanics. This begins in the
1930s with the polemics around the Hilbert space, leading to the creation of
quantum logic by von Neumann and Birkhoff. Up to its decay in the 1980s,
orthodox quantum logic has tried to apply discrete structures, such as orthomodular lattices, to capture the specificity of quantum theory. We explain
why these attempts failed. New methods appeared recently using categorytheoretic approaches. The focus of investigation shifted with the advent of
quantum information, from single systems to properties such as non-locality.
Entanglement and the rules of composition in quantum mechanics came to
be seen the key to quantumness. We describe the operational’ approach,
including methods based on convex sets, for understanding the amount of
non-locality and the Tsirelson bound. We particularly discuss a recent result
(Nature 461, 1101) called information causality’.
El Amin Kaidi
University of Almeria, Spain
Lecture: Espaces de Banach Hopfiens et Cohopfiens
Un espace de Banach X est dit Hopfien (respectivement Cohopfien) si tout
opérateur linéaire borné surjectif (respectivement injectif) est bijectif. L’existence
de tels espaces en dimension infinie n’a été résolue que récemment. Dans cet
exposé on va présenter quelques exemples, montrer une relation intéressante
avec le spectre descente pour les opérateurs d’un espace de Banach et les
éléments d’une algèbre de Banach. Des problèmes ouverts seront discutés.
Réferences:
A. Aviles and P. Kozmider, A Banach Space in which every injective operator is surjective. Bull. London Math. Soc.45 (2013) 1065-1074.
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M. Burgos, A. Kaidi, M. Mbekhta, and M. Oudghiri, The descent spectrum
and perturbations. J. Operator Theory 56 (2006) 259-271.
A. Haily, A. Kaidi, A. Rodrı́guez Palacios, Centralizers in semisimple algebras, and descent spectrum in Banach algebras. Journal of Algebra 347
(2011).
214-223.
A. Haily, A.Kaidi, A. Rodrı́guez Palacios. Algebra Descent Spectrum of
operators. Israel Journal of Mathematics 177 (2010), 349-368.
P.K Liau and C.K Liu, Invariants of algebraic derivations and automorphismes in Banach algebras. Preprint (2013).
Lajos Molnár
University of Debrecen, Hungary
Mini-course: Quantum structures and their transformations
We introduce some of the most fundamental quantum structures, namely,
structures consisting of states, effects, and (bounded) observables. In the
Hilbert space formalism of quantum theory these objects are represented
by density operators (i.e., positive Hilbert space operators with unit trace),
positive operators bounded by the identity, and bounded self-adjoint operators, respectively. We consider important algebraic operations, relations and
numerical quantities on their collections. Our main goal is the description of
symmetries, or, in other words, automorphisms of the so-obtain structures.
We discuss several classical results and also present some of our recent results
in particular on transformations of the set of states.
Markus Mueller
Heidlberg University, Germany
Mini-course: Quantum theory from five simple operational axioms
Quantum theory is usually formulated in terms of abstract mathematical
postulates, involving Hilbert spaces, complex wave functions etc.. But why
does nature work according to these counterintuitive rules? In this course, I
describe recent work on deriving quantum theory from simple, informationtheoretic postulates. This is similar to the usual formulation of special
relativity, where two simple physical principles - the principle of relativity
and the invariance of the speed of light - can be used as the starting point,
and the mathematics of Minkowski space can be derived from them.
First, I will introduce the framework of general probabilistic theories
(GPTs). This is a very simple mathematical framework, describing the basic
physical setting of having measurements that yield outcomes with certain
probabilities. I will show that quantum theory and classical probability are
special cases of GPTs, and I will discuss the theory of ”boxworld” which allows for ”superentangled” states that violate Bell inequalities by more than
any quantum state. Then, I introduce five simple axioms, and show how
quantum theory can be derived from them. We will first prove that a bit
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must have a state space that is a Euclidean ball, and then that it must be
three-dimensional, yielding the qubit and finally all higher-level quantum
state spaces.
In the final part of the course, I will discuss recent attempts to find
”quantum theory’s closest cousins”, that is, theories that are conceptually
closest to quantum theory and thus our best candidates of ”new physics”.
I will briefly describe experiments that have been performed to look for
those theories (such as experiments on 3rd-order interference), talk about
Jordan algebras, and introduce the quest for ”density-tensor” theories, an
important open problem in this context.
Peter Šemrl
University of Ljubljana, Slovenia
Mini-course: Symmetries on bounded observables and related
areas
In the mathematical foundations of quantum mechanics bounded observables are represented by bounded self-adjoint operators acting on a Hilbert
space H. The space of all bounded self-adjoint operators on H can be
equipped with various relations and operations that are important because
of their physical meaning. These are Jordan product, Jordan triple product, usual partial order, spectral order, and commutativity, among others.
Bijective maps on the space of all bounded self adjoint operators preserving
any of these operations or relations are called symmetries. They are studied
also on the subsets of positive operators, positive invertible operators and
the effect algebras. A new approach to the study of such maps based on
adjacency preserving maps will be presented. Connections with Geometry
and other areas of mathematical physics will be explained.
Barbara Terhal
Technical University, Aachen, Germany
Mini-course: Mathematical concepts in the theory of quantum
entanglement
In these lectures we will discuss several mathematical concepts which arise
in the theory of quantum entanglement for qubits. Topics include
1. Deciding separability: symmetric extensions of states and semi-definite
programming.
2. Bound entanglement and unextendible product bases.
3. Entanglement in fermionic systems: Gaussian states.
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Marı́a Victoria Velasco Collado
University of Granada, Spain
Mini-course: About the mathematical foundation of quantum
mechanics
In this course we give an overview about the Mathematical Foundation of
Quantum Mechanics. To this aim, we will give an idea about the main
problems that motivated the development of such subject. Then, we will
show why the development of the spectral theory became into an essential
tool for the mathematical substrate in which the Quantum Physics emerged.
Moreover we will review some basic references of the topic. Particularly we
will study the role of non-associative algebras in the origin of Quantum
Mechanic , and also in some new challenges for the further development of
the Modern Physic.
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Abstracts of talks
Hajar Benkhaldoun
University Moulay Ismail, Meknès, Morocco
On some applications of r-generalized Fibonacci sequences
The intent of this communication is to give the solution of the homogeneous difference equations with constant coeffcients. Our approach is based
on the r-generalized Fibonacci sequences. The tools used for proving this
(s)
result are the Casoratian and a family {Vn }n≥0 (0 ≤ s ≤ r − 1) of rgeneralized Fibonacci sequences.
Mustapha Boujeddaine
University Moulay Ismail, Meknès, Morocco
Fourier transforms of Dini-Lipschtz functions on rank 1
symmetric spaces
In this paper, we prove an analog of Younis’s theorem (version of Titchmarsh
theorem) on the image under the Fourier-Helgason transform of a set of
functions satisfying the Lipschitz functions in Lp (1 ¡ p 2) for functions on
noncompact rank 1 Riemannian symmetric spaces.
José Manuel Moreno Fernández
University of Malaga, Spain
The Gelfand-Kirillov dimension of graph algebras
We have studied features of the Gelfand-Kirillov dimension in the context
of non-necessarily unital algebras. These include the commutativity of the
GK-dimension with direct limits, its Morita invariance, its relationship with
rings of quotients... (See [2]).
For this talk, we have chosen to exhibit an application of the theory we have
developed in the context of graph algebras. We emphasize on Leavitt path
algebras, as these are the algebraic analog of graph C*-algebras, providing
the link between our work and this research school. Concretely, we will give
a theorem the determines the GK-dimension of graph algebras, extending
previously known results (See [1]).
Bibliography:
[1] A. Alahmadi, H. Alsulami, S.K. Jain, E. Zelmanov, Leavitt Path algebras
of finite Gelfand-Kirillov dimension, J. Algebra Appl. 11, no. 6, (2012)
1250225.
[2] J. M. Moreno Fernndez, M. Siles Molina, The Gelfand-Kirillov dimension
is Morita invariant for algebras with local units, preprint (2014).
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Abdelhadi Moutassim
CRMEF-Settat, Morocco
Operators of multiplication in absolute valued algebras satisfying
Le = R e
An absolute valued algebra is a non-zero real algebra that is equipped
with a multiplicative norm (kxyk = kxkkyk). We classify, by an algebraic
method, all four dimensional absolute valued algebras with operators of multiplication satisfying Le = Re where e is a non-zero idempotent. Moreover,
we give some new results in four dimensional absolute valued algebras with
left unit.
Hatim Naqos
University Mohammed V, Rabat, Morocco
Composition operators with univalent symbol in Schatten classes
We study composition operators, induced by a sub-domain of the unit disc
whose boundary intersects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation. We obtain an explicit characterization for the
membership in Schatten p-classes, in terms of γ.
This is a joint work with O. El-Fallah and M. El Ibbaoui.
Juana Sánchez Ortega
University of Málaga, Spain
Zero products preserving maps
The characterization of linear maps between Banach algebras, lattices or
spaces preserving certain properties like norm, spectrum, spectral radius,
commutativity, idempotents among many others, constitutes one of the most
thoroughly studied problems. (See, for example, [4, 5, 8, 9, 10, 11, 12, 13,
14]).
Let A and B be algebras. We say that a linear map T : A → B is zero
product preserving if T (a)T (b) = 0 whenever ab = 0. The question of
characterizing zero product preserving linear maps has been also considered
by a large number of authors [1, 2, 6, 7]. Araujo and Jarosz [3] showed
that every bijective linear map between unital standard operator algebras
which preserves zero product in both directions is a scalar multiple of an
algebra isomorphism. Shortly after that, zero product preserving maps were
described for prime rings containing non-trivial idempotents [6], and for ring
generated by idempotents [7].
This talk deals with zero product and Jordan zero product preserving maps
defined on a Banach algebra having enough minimal idempotents. To be
more precise, let A and B be unital complex algebras, assume that A is
semisimple with non-zero socle. Then, it can be proved that under certain
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conditions every bijective linear map T : A → B, which preserves zero
product, is an isomorphism multiplied by a central invertible element.
References
[1] J. Alaminos, M. Brešar, J. Extremera, and A.R. Villena, Maps preserving zero products, Studia Math. 193 (2009), 131–159.
[2] J. Alaminos, M. Brešar, J. Extremera, and A.R. Villena, Characterizing Jordan maps
on C*-algebras through zero products, Proc. Edinburgh Math. Soc. 53 (2010), 543–555.
[3] J. Araujo, K. Jarosz, Biseparating maps between operator algebras, J. Math. Anal.
Appl. 282 (2003), 48–55.
[4] M. Brešar, P. Semrl, Mappings which preserve idempotents, local automorphisms, and
local derivations, Canad. J. Math. 45 (3) (1993) 483–496.
[5] M. Brešar, P. Semrl, Linear preservers on B(X), in: Linear Operators, Warsaw, (1994),
in: Banach Center Publ., Vol. 38, Polish Acad. Science, Warsaw, 1997, 49–58.
[6] M. A. Chebotar, W. F. Ke, P. H. Lee, Maps characterized by action on zero products,
Pacific J. Math. 216 (2004), 217–228.
[7] M. A. Chebotar, W. F. Ke, P. H. Lee, N. C. Wong, Mappings preserving zero products,
Studia Math. 155 (2003), 77–94.
[8] A. Guterman, C. K. Li, P. Semrl, Some general techniques on linear preserver problems,
Linear Algebra Appl. 315 (13) (2000) 61–81.
[9] A. A. Jafarian, A. R. Sourour, Linear maps that preserve the commutant, double
commutant or the lattice of invariant subspaces, Linear Multilinear Algebra 38 (12)
(1994) 117–129.
[10] M. Marcus, B.N. Moyls, Linear transformations on algebras of matrices, Canad. J.
Math. 11 (1959) 61–66.
[11] L. Molnár, On isomorphisms of standard operator algebras, Studia Math. 142 (3)
(2000) 295–302.
[12] P. Semrl, Two characterizations of automorphisms on B(X), Studia Math. 105 (2)
(1993) 143–149.
[13] A. R. Sourour, Invertibility preserving linear maps on L(X), Trans. Amer. Math.
Soc. 348 (1) (1996) 13–30.
[14] M. Wolff, Disjointness preserving operators on C*-algebras, Arch. Math. (Basel) 62
(3) (1994) 248–253.
Hassan Zariouh
CRMEF-Oujda , Morocco
Preservation results for new Weyl-type theorems
The aim of this talk is to study the stability of extended Weyl and Browdertype Theorems (generalized or not) for orthogonal direct sum S ⊕ T, where
S and T are bounded linear operators acting on Banach spaces. We show in
particular that property (ab) in general is not preserved under direct sum.
Nonetheless, and under the assumptions that Π0a (T ) ⊂ σa (S) and Π0a (S) ⊂
σa (T ) we characterize the preservation of property (ab) under direct sum
S ⊕ T ; where Π0a (T ) is the set of left pole of T of finite rank. Furthermore we
show that if S and T satisfy generalized a-Browder’s theorem, then S ⊕ T
satisfies generalized a-Browder’s theorem if and only if σSBF − (S ⊕ T ) =
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σSBF − (S) ∪ σSBF − (T ), extending a recent result by removing certain extra
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assumptions; where σSBF − (T ) is the upper B-Weyl spectrum of T. The
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theory is exemplified in the case of some special classes of operators.
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References
[1] M. Berkani, H. Zariouh, New extended Weyl type theorems, Mat. Vesnik 62(2),
145–154, 2010.
[2] M. Berkani, M. Kachad, H. Zariouh, Extended Weyl-type Theorems for direct
sums, Demonstratio Mathematica, to appear.
Posters on Quantum information by Phd students from
University Mohammed V, Rabat
Hanane El Hadfi, ” Geometric quantum discord through the schatteen
1-norm”.
Rim Essaber, ” The evolution of geometric quantum discord of two
qubits in independent reservoirs”.
Wiam Kaydi, ” Monogamy of geometric quantum discord in photonadded coherent states” .
Sanaa Seddik, ”Communication via non-markovian and markovian dynamics under noisy environment.”