Author: Ngọc Nhi Nguyễn

30/10/2024 : Clotilde Fermanian Kammerer @PoliMi

Ngọc Nhi NguyễnSeminar PoliMi 
Wednesday, October 30, 2024 - 14:00

Sala Consiglio (7th floor), D-Mat
Mathematics Department of PoliMi
Campus Leonardo, bd.14 "Nave".

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SPEAKER: Clotilde Fermanian Kammerer (Université d’Angers)

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Semi-classical measures, two scale semi-classical measures and applications

In this lecture, we will present semi-classical measures and show how they  describe the obstructions to strong convergence of bounded families of square integrable functions. We will also describe applications for families of solutions to PDEs, in particular to wave equations.


This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of    Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

9 &10/10/2024 : Ludovico Lami @PoliMi

Ngọc Nhi NguyễnSeminar PoliMi 
Wednesday, October 9, 2024 - 15:00
Thursday, October 10, 2024 - 11:30

Sala Consiglio (7th floor), D-Mat
Mathematics Department of PoliMi
Campus Leonardo, bd.14 "Nave".

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SPEAKER: Ludovico Lami (University of Amsterdam)

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A quantitative approach to entanglement theory via hypothesis testing (Oct. 9)

I will start by presenting the notion of entanglement as studied in quantum information theory. According to this definition, originally proposed by Werner in 1989, a density operator on a bipartite quantum system is declared to be entangled if it cannot be written as a convex combination of tensor products of single-system density operators, and separable (or unentangled) otherwise. I will then discuss the basics of quantum hypothesis testing and introduce the task of “entanglement testing”, which consists in discriminating a given entangled state from the set of all separable states. This task is a fundamental quantum information primitive, with applications ranging from device certification to gravitational entanglement detection. I will finish by discussing the statement of the “generalised quantum Stein’s lemma”, which connects the ultimate efficiency of entanglement testing with a key entanglement measure known as “relative entropy of entanglement”.

A solution of the generalised quantum Stein’s lemma (Oct. 10)


I will discuss the solution of the generalised quantum Stein’s lemma presented in [Lami, arXiv:2408.06410] (see also [Hayashi/Yamasaki, arXiv:2408.02722] forrelated work), which establishes that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing between n copies of an entangled state and a generic separable state, equals the regularised relative entropy of entanglement. To solve the problem I will briefly introduce two techniques. The first is a procedure called “blurring”, which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. I will discuss this technique extensively in the classical case, where it already suffices to prove the generalised Stein’s lemma. Depending on time, I will then present a second technical innovation, which is needed to prove the quantum version of the statement. This consists in a second quantisation step, which lifts the problem from a finite-dimensional system to an infinite-dimensionalbosonic quantum system, where it can then be solved with techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel.


This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

05/06/2024 : Umberto Morellini @UniMi

Ngọc Nhi NguyễnSeminar UniMi 
Wednesday, June 5, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Umberto Morellini (Université Paris Dauphine, CEREMADE, PSL) ________________________________________________________________________

The free energy of Dirac’s vacuum in purely magnetic fields

The Dirac vacuum is a non-linear polarisable medium rather than an empty space. This non-linear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at non-zero temperature needs to be better understood.
In this talk, we will present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of quantum vacuum in a classical magnetic field. After introducing our model, we will properly define the free energy functional using the Pauli-Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well known issue of the theory. The study of the properties of this functional will be addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg
formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in 1979. The talk is based on the work available at arXiv:2404.12733.

22/04/2024: Antoine Prouff @UniMi

Ngọc Nhi NguyễnSeminar UniMi 
Monday, April 22, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Antoine Prouff (Université Paris-Saclay, Laboratoire de Mathématique d’Orsay) _________________________________________________________________________

Egorov’s theorem in the Weyl-Hörmander calculus and application to the control of PDEs

It is known that geometric optics can be derived as the high-frequency limit of the wave equation, from both experimental and theoretical perspectives. This fact can be regarded as an instance of a “quantum-classical correspondence principle”, made rigorous by Egorov’s theorem, which relates the evolution of a linear PDE (e.g. the wave equation) to the natural underlying classical dynamics (e.g. the geodesic flow).

We will present a version of Egorov’s theorem in the Euclidean space, in the setting of the “Weyl-Hörmander calculus”. This general framework of microlocal analysis involves Riemannian metrics on the phase space adapted to the dynamics under consideration, and allows for a fairly large range of applications (study of Schrödinger, wave and transport equations).

If time allows, we will discuss in more detail an application to the observability of the Schrödinger equation with a confining potential in the Euclidean space.