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a joint seminar of the IMT and the LPT

par

, - 23 octobre 2020

The mathematical physics seminar is organized by mathematicians from the IMT and theoretical physicists from the LPT on a bi-weekly basis. The talks usually take place Thursdays at 14:00.

For more information, contact one of the organizers : Tristan Benoist, Cécilia Lancien, Ion Nechita or Clément Pellegrini.

- [Thu June 17th 2021, 14:00, Salle Katherine Johnson (IMT) and online] Naïmo Davier (LPT) -
*GdT Effet Hall II*We have already explored the physics of integer quantum all effect, explaining why we observe plateaux in the transverse resistivity labeled by integers. This is not all the story, indeed if one look closer there is other smaller plateaux appearing, especially for very strong magnetic field. In order to explain this observation one need to take interactions between electrons into account. We will start the analyse with the introduction of many particles Laughlin wave function, and we will see excitations of this new state are creation of quasi-particles. It can actually be showned that this two dimensional particles are anyons, wich will be a good occasion to introduce this type of particles. Depending on time remaining we could talk briefly about composite fermions as complementary model explaining the observed behaviour for transverse resistivity.

- [Thu June 24th 2021, 14:00, online] Cambyse Rouzé (TU Munich) -
*TBA*

- [Thu June 3rd 2021, 14:00, Salle Katherine Johnson (IMT) and online] Clément Tauber (IRMA, Strasbourg) -
*Topological indices for shallow-water waves*In this talk, I will apply tools from topological insulators to a fluid dynamics problem : the rotating shallow-water wave model with odd viscosity. The bulk-edge correspondence explains the presence of remarkably stable waves propagating towards the east along the equator and observed in some Earth oceanic layers. The odd viscous term is a small-scale regularization that provides a well defined Chern number for this continuous model where momentum space is unbounded. Equatorial waves then appear as interface modes between two hemispheres with a different topology. However, in presence of a sharp boundary there is a surprising mismatch in the bulk-edge correspondence : the number of edge modes depends on the boundary condition. I will explain the origin of such a mismatch using scattering theory and Levinson’s theorem. This talk is based on a series of joint works with Pierre Delplace, Antoine Venaille, Gian Michele Graf and Hansueli Jud.

- [Thu May 20th 2021, 14:00, online] Naïmo Davier (LPT) -
*GdT Effet Hall I* - [Thu April 29th 2021, 14:00, online] Manon Michel (LMBP, Université Clermont-Auvergne) -
*Numerical and theoretical unification of graph representation of the Potts model*The q-state Potts model plays a major role in the study of critical phenomena, mostly thanks to the diff-erent representations one can use for its description. Historically, a coupling between the spin and Fortuin-Kasteleyn (FK) representations was derived. It was successfully and extensively used for theoretical as well as numerical purposes for forty years. On another hand, the loop representation, generalized into the q-flow one for general q, has proven to be important for the study of the magnetic susceptibility or correlation length, thanks to the worm algorithm. However, in the absence of any coupling to any of the other representations, the loop representation still has to be considered independently. Recently, Grimmett and Janson (2009) showed how to build on the properties of random even graphs to build a coupling between the FK and loop representations for the Ising model. During this talk, I will explain how a more physical approach based on the decomposition of random clusters into loop ones completes the construction of this missing link and ties together the three representations by the introduction of a joint-model between the FK and loop ones. This model allows a direct coupling between both these representations, recovering the special case of the Ising model developed by Grimmett and Janson (2009).

- [Thu April 15th 2021, 14:00, online] Antoine Tilloy (MPQ & MCQST Munich) -
*Relativistic continuous matrix product states for quantum fields without cutoff*The variational method is a powerful approach to solve many-body quantum problems non perturbatively. However, in the context of relativistic quantum field theory (QFT), it needs to meet 3 seemingly incompatible requirements outlined by Feynman : extensivity, computability, and lack of UV sensitivity. In practice, variational methods break one of the 3, which translates into the need to have an IR or UV cutoff. In this talk, I will introduce a relativistic modification of continuous matrix product states that satisfies the 3 requirements jointly in 1+1 dimensions. I will then show how to apply the method to the self-interacting scalar field, without UV cutoff and directly in the thermodynamic limit.

- [Thu March 18th 2021, 14:00, online] Simon Apers (CWI Amsterdam & ULB Brussels) -
*Quantum walks and electric networks*Quantum walks on graphs are the quantum analogue of random walks on graphs. They are a key primitive in quantum computing, leading to faster quantum algorithms for problems in combinatorial optimization and property testing. In this talk I will describe the close connection between quantum walks and electric networks. Such a connection also exists between random walks and electric networks, and it has been extremely useful. As an illustration I will show how this more recent connection leads to quantum walk algorithms for solving Laplacian systems, estimating random walk quantities and speeding up the hitting time.

- [Thu February 18th 2021, 14:00, online] Lea Bossmann -
*Asymptotic expansion of low-energy excitations for weakly interacting bosons*We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N. Joint work with Sören Petrat and Robert Seiringer.

- [Thu February 4th 2021, 14:00, onsite and online] Nicolas Rougerie -
*Semi-classical limit for almost fermionic anyons*Fundamental particles come in two types : fermions and bosons, according to whether they satisfy the Pauli exclusion principle or they do not. However, quasi-particles of certain low-dimensional condensed matter systems may violate this fundamental dichotomy and have an intermediate behavior. Such exotic objects, called anyons, can be described as ordinary bosons or fermions with special long-range magnetic interactions. This leads to intricate models for which well-educated approximations are desirable.

In this talk we study a limit situation where the anyon statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. Our proof is based on coherent states, Husimi functions, the Diaconis-Freedman theorem and a quantitative version of a semi-classical Pauli pinciple.

Joint work with Théotime Girardot. - [Thu January 21st 2021, 14:00, online] Cédric Bernardin -
*Spiking and collapsing in quantum trajectories*Motivated by recent studies by Bauer et al., we consider strong noise limit of some (matrix valued) stochastic differential equations called Belavkin equations, arising from quantum measurements. As the noise (i.e. measurement) grows larger, the solutions exhibits locally a collapsing, that is to say converge to jump processes, very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. In this talk we will present rigorous results about the convergence of the Belavkin equations to the jump process and in the case of a qubit we will show how to obtain also the spike process decorating the underlying jump process. Joint work with T. Benoist, R. Chhaibi, R. Chetrite, J. Najnudel and C. Pellegrini

- [Thu December 17th 2020, 14:00, online] Reda Chhaibi (IMT) -
*Group homologies*[notes] - [Thu December 10th 2020, 14:00, online] Leevi Leppäjärvi (Turku University) -
*Simulation of measurement devices : an overview*[slides]The simulation scheme describes a process of obtaining new measurement devices out of some existing ones by the means of operational manipulations of mixing and post-processing. We consider this simulation of measurement devices within the operational framework of general probabilistic theories (GPTs) and see how it relates to several different concepts in GPTs and quantum theory. First, we introduce the simulation irreducible observables that can only be simulated by itself but which can be used to simulate any other observable. We use the simulation irreducible observables to characterize the set of fully compatible observables that can be measured jointly with any other observable. We find that there are theories where the so-called no-free-information principle does not hold meaning that, unlike in quantum theory, the set of fully compatible observables does not coincide with the trivial coin-tossing observables. On the other hand, we argue that if one is to go beyond the no-restriction hypothesis where not all mathematically valid effects or observables are taken to be measurements in the theory, the physically feasible set of measurements should still be closed under simulation. Furthermore, we give examples of restrictions of measurements of different types. Lastly, we formalize the concept of post-processing and simulation of quantum instruments and see how it relates to post-processing and simulation of POVMs in quantum theory.

Main references :

S. N. Filippov, T. Heinosaari and L. Leppäjärvi, Simulability of observables in general probabilistic theories, Phys. Rev. A 97, 062102 (2018).

T. Heinosaari, L. Leppäjärvi, M. Plávala, No-free-information principle in general probabilistic theories, Quantum 3, 157 (2019).

S. N. Filippov, S. Gudder, T. Heinosaari, L. Leppäjärvi, Operational restrictions in general probabilistic theories, Found. Phys. 50, 850-876 (2020).

L. Leppäjärvi and M. Sedlák, Post-processing of quantum instrument, arXiv:2010.15816 (2020). - [Thu November 26th 2020, 17:00, online] Martin Fraas (Virginia Tech) -
*Quantized quantum transport in interacting systems*For non-interacting fermions at zero temperature, it is well established that charge transport is quantized whenever the chemical potential lies in a gap of the single-body Hamiltonian. Proving the same result with interactions was an open problem for nearly 30 years until it was solved a few years by Hastings and Michalakis. The solution uses new tools originally developed in the context of the classification of exotic phases of matter, and was used before in the proof of the many-dimensional Lieb-Schultz-Mattis theorem. I will explain these developments and show a theorem that unifies most of the known results on the subject. The talk is based on a joint work with Alex Bols, Sven Bachmann and Wojciech De Roeck.

- [Thu November 19th 2020, 14:00, online] Tristan Benoist (IMT) -
*On Schuch et al. "Classifying quantum phases using Matrix Product States and PEPS", arXiv:1010.3732* - [Thu October 22nd 2020, 14:00, online] Anna Francuz (Jagiellonian University) -
*Determining topological order from infinite projected entangled pair states*In my talk, after a short introduction to topological order, I will present a method of extracting information about the topological phase starting from a ground state of a lattice Hamiltonian. Using the tensor network formalism I will show how to obtain topological S and T matrices, encoding the mutual statistics and self-statistics of emergent anyons. In most cases they determine the topological phase unequivocally and in that sense they can be thought of as some kind of non-local order parameter of topological phases of matter. I will show that with 2D tensor network - PEPS, the method allows to analise states which were not achievable by 2D DMRG due to long correlation length. In the end I will show results related to determining topological order for non Abelian anyon models and a way of extracting topological entanglement entropy directly in the thermodynamic limit.

- [Thu October 8th 2020, 14:00, IMT, bat 1R1, salle 106, and online] Faedi Loulidi (LPT) -
*GdT Quantum Spin Systems (X) : 1D spin system and their classification with the Cohomology theory*In this presentation and will try to give an explicit calculation of 1D spin chain with site symmetry and the "emergence" of the Group Cohomology where in 1d is just the projective representation of the Group that classify different phase of matter. This will motivate us to give an introduction of Group Cohomology in d dimension and it’s fundamental role in topological phases for further exploration(s).

Some references that I used for my talk :

https://arxiv.org/abs/1010.3732

https://arxiv.org/abs/1008.3745

An introduction to the subject in this lecture notes : http://topo-houches.pks.mpg.de/wp-content/uploads/2015/01/pollmann_spt.pdf

And a mathematical (comprehensible) introduction to Group Cohomology of a Group : https://alistairsavage.ca/pubs/Mendonca-Projective_Representation_of_Groups.pdf - [Thu September 24th 2020, 14:00, online] András Molnár (Complutense University of Madrid) -
*Fundamental theorems for PEPS*[slides]Tensor networks are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one : different sets of tensors may generate the very same state. A fundamental question in the study of tensor networks naturally arises : what is the relation between those different sets, if there is one at all ? The answer to this question in one dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. In this talk I explore what are the possible answers to this question for projected entangled-pair states (PEPS) in two- and more dimensions.

- [Thu September 17th 2020, 14:00, IMT, bat 1R1, salle 106] Faedi Loulidi (LPT) -
*GdT Quantum Spin Systems (IX) : Classiﬁcation of gapped phases in one-dimensional spin systems*References :

- Classification of Gapped Symmetric Phases in 1D Spin Systems (Wen et al)
- Renormalisation -Group Transformations on Quantum states ( Verstraete, Cirac, Latorre, Rico, Wolf)
- Matrix Product State Representations (Perez Garcia, Verstraete, Wolf,Cirac)
- Chapter 8 of the book Quantum information meet Quantum matter
- Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order (Xie Chen, Zheng-Cheng Gu,Xiao-Gang Wen)

- [Thu July 9th 2020, 14:00] Amanda Young (TU München) -
*Methods for estimating spectral gaps in quantum spin systems : Applications to AKLT and ν=1/3 fractional quantum Hall systems*The existence of a non-vanishing spectral gap above the ground state energy is a central question in the study of quantum spin systems. We will discuss approaches for establishing non-zero spectral gaps for frustration-free models, and illustrate these through two applications : first, to a two-dimensional decorated AKLT model [1] and second, to a ν=1/3 fractional quantum Hall (FQH) model [2,3]. The methods used to estimate these gaps are based on estimates involving ground state projections, and so having a good description of the ground state space is key. For the decorated AKLT model this is given by Tensor Network States defined using the MPS for the one-dimensional AKLT model, while the ground states of the FQH model are best described using lattice tilings that produce states we refer to as fragmented matrix product states. Key properties of both ground state representations will be discussed. This talk is based off the follow work : [1] Abdul-Rahman, H., Lemm, M., Lucia, A., Nachtergaele, B. and Young A., A class of two-dimensional AKLT models with a gap, in Analytic Trends in Mathematical Physics, Houssam Abdul-Rahman, Robert Sims, Amanda Young (Eds), Contemporary Mathematics vol 741, pp 1-21 (2020), American Mathematical Society [2] Nachtergaele, B., Warzel, S. and Young, A., Spectral Gaps and Incompressibility in a ν=1/3 Fractional Quantum Hall System, arXiv:2004.04992 (2020) [3] Nachtergaele, B., Warzel, S. and Young, A., Low-complexity eigenstates of a ν=1/3 fractional quantum Hall system, arXiv:2006.00300 (2020)

- [Thu June 18th 2020, 14:00] Cécilia Lancien (IMT) -
*GdT Quantum Spin Systems (IX) : Parent Hamiltonians of tensor network states* - [Wed June 10th 2020, 14:00] Denis Rochette (IMT&LPT) -
*Optimal cloning of pure states*Quantum cloning is impossible according to the no-cloning theorem, but it is still possible to perform imperfect quantum cloning and to give explicit error bounds. We shall focus on the M to N symmetric cloning of pure states where the copies live in the symmetric subspace.

- [Thu May 28th 2020, 14:00] Olivier Gauthé (EPFL) -
*GdT Quantum Spin Systems (VIII) : Tensor networks for thermal states*Tensor network are known to be efficient representation of low-energy quantum states in 1D thanks to the area law of entanglement. In 2D, less exact results are known, but empiric data show they provide a good ansatz for those states. However, they cannot be used to represent high-energy states that are typically highly entangled. Here, we review how tensor network can be used to efficiently represent thermal density matrices at any temperature, relying on the concept of purification and enlarged Hilbert space. We will see that the area law still holds and exact bounds are known for virtual dimension. References : Verstraete, Garcia-Ripoll and Cirac, PRL 93 (2004) Wolf, Verstraete, Hastings and Cirac , PRL 100 , (2008) Molnar, Schuch, Verstraete and Cirac, , PRB 91 , (2015)

- [Thu May 14th 2020, 14:00 ; BBB link] Juraj Hasik (LPT) -
*GdT Quantum Spin Systems (VII) : Injective and non-injective MPS* - [Thu April 30th, 14:00] Elba Garcia-Failde (IPhT Paris-Saclay) -
*Topological recursion and the O(n) loop model*In this talk we will further study the topological recursion as a process to go from the base topologies of disks and cylinders to arbitrary topologies. In particular, we will see how to make use of it to explore the O(n) loop model, a statistical model which we consider as an ensemble of decorations on random maps. The goal is to describe the critical behavior of the generating series of maps endowed with loops in the so-called dense and dilute phases when the volume, which is the number of vertices, is large and considering the maps to have a prescribed number of large and small boundaries. This allows to investigate the nesting properties of loops by associating to every map with a loop configuration a graph which encodes all the relevant nesting information and checking which graphs are more likely to occur, which leads to interesting qualitative conclusions about the most probable shapes of maps in the model.

- [Thu April 23rd, 14:00] Stephane Dartois (LaBRI, Bordeaux) -
*Introduction to the Eynard-Orantin Topological Recursion*[slides]In this talk I will introduce the topological recursion formula and show how it is obtained in the classical case of the Hermitian matrix model starting from loop equations. After this is done I will show how it concretely works on specific examples drawn both from the world of matrix models and outside this world.

- [Thu April 16th 2020, 14:00] Ion Nechita (LPT) -
*GdT Quantum Spin Systems (VI) : Approximating ground states of gapped 1D Hamiltonians by Matrix Product States*[notes]I will discuss Section 2 of [Hastings - An area law for one-dimensional quantum systems], showing how to use the entropy bounds that Tristan derived the last time to approximate the ground state of gapped 1D quantum systems by matrix product states. I will also present some related results (e.g. other Renyi entropies) from [Verstraete, Cirac - Matrix product states represent ground states faithfully] and [Schuch et al - Entropy Scaling and Simulability by Matrix Product States].

- [Thu March 26th 2020, 10:00] Tristan Benoist (IMT) -
*GdT Quantum Spin Systems (V) : Hastings’ proof of the area law for gapped Hamiltonians* - [Thu March 12th 2020, 14:00 ; room 106, 1R1, IMT] Yan Pautrat (LMO, Paris-Saclay) -
*Variation de la chaleur dans les systèmes quantiques*Considérons un système infini et initialement à l’équilibre. Si l’on perturbe sa dynamique, il n’est plus à l’équilibre et va donc évoluer ; on s’intéresse à la variation de chaleur lors de cette évolution. Pour un système classique, il est immédiat que cette variation est bornée si la perturbation l’est. Pour un système quantique, le problème est bien plus compliqué et les difficultés commencent au niveau de la définition de la variation. Une première définition, naïve, n’est pas acceptable physiquement ; une autre définition, par des mesures à deux temps, est plus satisfaisante mais nous verrons que la variation de chaleur ainsi définie est une variable aléatoire qui peut présenter des variations non bornées, à moins qu’une certaine condition ultraviolette ne soit vérifiée. Nous présenterons les conséquences de cette condition, en particulier quand le système initial est constitué de plusieurs réservoirs initialement à l’équilibre thermique. Nous discuterons en particulier une traduction du premier principe de la thermodynamique au niveau des lois des variations de chaleur des différents réservoirs.

- [Thu February 6th 2020, 14:00 ; room 106, 1R1, IMT] Ion Nechita (LPT) -
*GdT Quantum Spin Systems (IV) : Lieb-Robinson bounds*[ notes ] - [Thu January 23rd 2020, 14:00 ; room 106, 1R1, IMT] Yassir Dehmane and Faedi Loulidi (LPT) -
*GdT Quantum Spin Systems (III) : Thermodynamic limit and KMS states* - [Thu January 9th 2020, 14:00 ; room 106, 1R1, IMT] Yassir Dehmane and Faedi Loulidi (LPT) -
*GdT Quantum Spin Systems (II) : Thermodynamic limit and KMS states*Given a lattice system, describing physical model with a "microscopic" Hamiltonian generating dynamics. In such models eg:Heisenberg spin chain,X-Y model ... interesting phenomena may occur such as phase transition in the thermodynamic limit The main object of this first lecture to answer these questions : Does the dynamics always exist even in the thermodynamic limit ? How could we characterise different phase transition ?

- [Thu December 12th 2019, 14:00 ; room 20, ground floor, LPT] Gemma DE LAS CUEVAS (Universität Innsbruck) -
*Local descriptions of mixed states*[ slides ]Local descriptions of mixed states face a fundamental challenge, namely, one cannot have a description which is both efficient and locally positive. This problem is due to a connection to nonnegative matrices, and the fact that there is a separation between the rank and positive semidefinite rank. More generally, I will explain how several decompositions of mixed states reduce to decompositions of nonnegative matrices such as the nonnegative factorisation, the positive semidefinite factorisation, the completely positive factorisation and the completely positive semidefinite decomposition. I will also explain a recent connection to free spectrahedra. Mostly based on recent joint work with Tim Netzer.

- [Thu November 28th 2019, 14:00 ; room 106, bat 1R1, IMT] Juraj Hasik (LPT) -
*GdT Quantum Spin Systems (I)*The spin models arise in the condensed matter physics as an effective description of insulating materials. They serve as a main tool in modelling and understanding the magnetic phenomena in Nature. In the first session of GdT on quantum spin systems, I will introduce the basic notions - Hilbert space of a spin system, its algebra of observables and the quantum dynamics. I will further illustrate these concepts on a paradigmatic spin model - Heisenberg chain.

- [Thu September 19th 2019, 14:00 ; room 20, ground floor, LPT] Motohisa FUKUDA (Yamagata University) -
*Miscellaneous topics in deep learning*[ photo | slides ]Recently, I started developing theoretical and practical research projects in the field of deep learning. In this talk, I plan to share some of what I have learnt. In the first part of this talk I introduce what deep learning is like, and then in the second I go over the initialization problem of deep neural networks, where random matrix theory can play an important role. Finally, I show some examples of image segmentation, which were obtained in the on-going joint project of applying deep learning to agricultural science.

- [Thu June 27th 2019, 15:00 ; Salle de Séminaire IRSAMC, portes N.325 et 327- 3e étage] Maria JIVULESCU (Politehnica, Timisoara) -
*Random positive operator valued measures*[ photo | slides ]We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue distributions with the help of free probability theory. Similarly, we obtain the large system limit for several quantities of interest in quantum information theory, such as the sharpness, the noise content, and the probability range. Finally, we study different compatibility criteria, and we compare them for generic POVMs. Joint work with Teiko Heinosaari and I. Nechita, see arXiv preprint.

- [Thu June 13th 2019, 14:00 ; Salle MIP, IMT] Tony JIN (LPTENS, Paris) -
*The quantum symmetric simple exclusion process*[ photo | slides ]I will present and discuss some results concerning a model of fermions hopping between neighboring sites on a line with random Brownian amplitudes. Both the periodic case and the open case will be discussed. In mean, such model maps to classical exclusion processes although the quantum nature of the model reveals itself in the non vanishing steady state fluctuations. Such fluctuations abide by a large deviation principle for large system size. For the closed case, the Harish-Chandra-Itzykson-Zuber integral is shown to play a predominant yet unexpected role in the derivation of the steady-state measure. For the open case, I will present a systematic recursive way using diagrammatic tools to compute exactly the large-deviation function at each order.

- [Mon May 27th 2019, 14:00 ; Salle de Séminaire IRSAMC, portes N.325 et 327- 3e étage] Olivier GIRAUD (LPTMS, Orsay) -
*Entanglement and the truncated moment problem*We present a tensor representation for spin states that finds applications in the characterization of the convex hull of spin coherent states. By making a connection between this problem an the truncated moment sequence problem, we develop an algorithm for the detection of entanglement in symmetric states.

- [Wed April 10th 2019, 11:00 ; Salle 106, IMT] Christian MAJENZ (QuSoft, University of Amsterdam) -
*The mathematical structure of port-based teleportation*Port-based teleportation is a variant of the ubiquitous quantum teleportation protocol by Bennett et al., where the receiver has to merely pick up the right quantum system, called “port”, instead of having to apply a unitary correction. While much less efficient and fundamentally imperfect, it enables applications such as instantaneous non-local computation due to a “simultaneous unitary covariance” property. After introducing the symmetries of the problem, that enable an application of Schur-Weyl duality, I will show how the leading order asymptotics for large number of ports of the optimal error is related to the principal eigenvalue of the Dirichlet Laplacian on the simplex of ordered probability distributions. If time permits, I will briefly explain a strengthening of a convergence result for a family of probability distributions arising in representation theory and combinatorics that can be used to analyze the asymptotics of the simpler standard protocol.

- [Thu March 14th 2019, 14:00 ; Salle 106, IMT] Pierre PUJOL (LPT, Toulouse) -
*Etats topologiques de la matière, exemples et classification* - [Thu February 21st 2019, 14:00 ; Salle 106, IMT] Pierre PUJOL (LPT, Toulouse) -
*Théorie conformes : champ scalaire, et fonction de partition sur le tore du champ compactifié* - [Thu January 31st 2019, 14:00 ; Salle 106, IMT] Francesco COSTANTINO (IMT, Toulouse) -
*Une introduction aux théories conformes des champs selon Segal (suite)* - [Thu January 24th 2019, 14:00 ; Salle 106, IMT] Francesco COSTANTINO (IMT, Toulouse) -
*Une introduction aux théories conformes des champs selon Segal* - [Tue December 18th 2018, 14:00 ; Salle 106, IMT] Carlos GONZALEZ-GUILLEN (UPM, Madrid) -
*On the spectral gap of random quantum channels*[ photo ]We prove a lower bound on the difference between the first and second singular values of quantum channels induced by random isometries, that is tight in the scaling of the number of Kraus operators. This allows us to give an upper bound on the difference between the first and second largest (in modulus) eigenvalues of random channels with same large input and output dimensions for finite number of Kraus operators k≥169. Moreover, we show that these random quantum channels are quantum expanders, answering a question posed by Hastings. As an application, we show that ground states of infinite 1D spin chains, which are well-approximated by matrix product states, fulfill a principle of maximum entropy.

- [Thu November 29nd 2018, 14:00] Simon SCHMIDT (Universität des Saarlandes) -
*Quantum automorphism groups of graphs*[ photo ]To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowicz’s compact matrix quantum groups. The first part of the talk will concern compact matrix quantum groups. As an important example we discuss the quantum symmetric group, the quantum analogue of the symmetric group. Quantum automorphism groups of graphs are certain quantum subgroups of the quantum symmetric group. In the second part we will look at the Petersen graph and see that this graph does not have quantum symmetry, i.e. its quantum automorphism group is commutative.

- [Thu November 22nd 2018, 14:00] Andreas BLUHM (TU Munich) -
*Compatibility of quantum effects and inclusion of free spectrahedra : the details*[ photo | slides ]One of the defining properties of quantum mechanics is the existence of incompatible observables, of which the observables of position and momentum are a well-known example. In this talk, we will connect the problem of determining whether a given set of measurements is compatible to the inclusion of free spectrahedra. Free spectrahedra are objects arising in convex optimization. After a short introduction to the theory of free spectrahedra, we show how results from algebraic convexity can be used to quantify the degree of incompatibility of binary quantum measurements. In particular, this new connection will allow to completely characterize the case in which the dimension of the quantum system is exponential in the number of measurements.