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Upcoming Talks
- Title: The NPA hierarchy does not necessarily attain the commuting operator value
Speaker: William Slofstra
Date and time: Monday, September 8, at 14h00
Location: Alan Turing Building – Amphithéâtre Sophie Germain
Abstract: Given a nonlocal game, we’d like to find the quantum value, which is the optimal winning probability with entanglement. The standard method for computing the value is to use the NPA hierarchy, which provides a sequence of semidefinite programs (SDPs) which upper bound the quantum value. These upper bounds decrease as the level of the hierarchy increases, and converge to the commuting operator value of the game, which is the optimal winning probability in the commuting operator model of entanglement. Thanks to the celebrated MIP=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen, we now know that the commuting operator value of the game can be strictly larger than the quantum value, so the NPA hierarchy cannot be used to compute the quantum value in general. In fact, the MIP=RE theorem states that the quantum value is uncomputable (even approximately). The MIP^{co} = coRE conjecture states that the commuting operator value is similarly uncomputable. Nonetheless, the NPA hierarchy is widely used. Looking at the decreasing sequence of values, it’s natural to wonder: is there always a level of the hierarchy that attains the commuting operator value. Somewhat surprisingly, the the MIP*=RE theorem and the MIP^{co}=coRE conjecture do not seem to answer this question (they just imply that if the value is always attained, then the level where the value is attained must be an uncomputable function of the game). In this talk, I’ll cover joint work with Marco Fanizza, Larissa Kroell, Arthur Mehta, Connor Paddock, Denis Rochette, and Yuming Zhao, where we resolve this question in the negative via a different computability result: we show that it’s undecidable to determine whether the value of the game is strictly greater than 1/2. I’ll also explain why this computability result doesn’t seem to follow from MIP*=RE.
Video: link
- Title: Algorithms for quantum thermal states
Speaker: Hamza Fawzi
Date and time: Monday, September 1, at 14h00
Location: Alan Turing Building – Amphithéâtre Sophie Germain
Abstract: Computing properties of thermal states is a central yet notoriously hard question in quantum many-body systems. In this talk I will discuss classical algorithms for (quantum) Gibbs states, including the computation of observables and partition functions. Our algorithms make use of quantum relative entropy optimization and provide rigorous guarantees on the quantities of interest.
Video: link
Past Talks
- Title: Certified Many-Body Physics
Speaker: Antonio Acin
Date and time: Thursday, May 22, 2025, at 14h00
Location: CPHT, Ecole Polytechnique, Room Luis Michel
Abstract: When studying many-body systems, two approaches have been considered so far: analytical derivations and variational methods. The first provide exact results, as they do not involve any approximations, but scale exponentially with the number of particles, while the second scale much better but only provide estimates with no theoretical guarantees. Polynomial optimisation methods offer an alternative approach somehow combining the advantage of exact and variational methods: it provides rigorous results, now in the form of upper and lower bounds, in a scalable way. We illustrate this new approach in two paradigmatic many-body problems: the estimation of expectation values in ground states of Hamiltonian operators and in steady states of quantum open systems.
Video: link
- Title: The Navascués, Pironio, Acín (NPA) hierarchy and its applications to condensed matter
Speaker: Igor Klep
Date and time: Thursday, April 24, 2025, at 14h00
Location: CPHT, Ecole Polytechnique, Room Luis Michel
Abstract: Noncommutative polynomial optimization (NPO) deals with optimization problems involving polynomial functions of variables that do not commute, typically represented as operators or matrices. Such problems naturally arise in condensed matter physics (e.g., estimating ground state energies of quantum many-body systems), quantum information theory (e.g., bounding entanglement measures), quantum chemistry (e.g., approximating molecular energy levels), and operator theory (e.g., solving noncommutative problems involving matrix inequalities).
In this talk, I shall present the Navascués-Pironio-Acín (NPA) hierarchy, a powerful method that provides a converging sequence of semidefinite programming (SDP) relaxations for solving NPO problems. By encoding moment constraints of noncommuting operators, the NPA hierarchy yields tractable and systematically improvable bounds, with convergence guarantees to the true optimum. This makes it a valuable tool for tackling optimization problems in quantum physics and beyond.
Video: link
- Title: Approximations for polynomial optimization on the sphere and quantum de Finetti theorems
Speaker: Monique Laurent
Date and time: Monday, April 7, 2025, at 14h15
Location: Alan Turing Building – Amphithéâtre Sophie Germain
Abstract: We revisit two approximation hierarchies for polynomial optimization on the unit sphere, whose convergence analysis for the r-th level bound was shown to be, respectively, in O(1/rˆ2) by Fang and Fawzi (in 2020, using the polynomial kernel method) and in O(1/r) by Lovitz and Johnston (in 2023, using a quantum de Finetti theorem of Christandl et al. (2007) for complex matrices with Bose symmetry).
We investigate links between these approaches, in particular, via duality of moments and sums of squares. In particular, we propose another proof for the analysis of the spectral bounds of Lovitz and Johnston, via a “banded” real de Finetti theorem for real Bose symmetric matrices, and we show that the spectral bounds cannot have a convergence rate better than O(1/rˆ2). In addition, we show how to use the polynomial kernel method to obtain a de Finetti type result for real maximally symmetric matrices, improving an earlier result of Doherty and Wehner (2013).
Joint work with Alexander Taveira Blomenhofer, University of Copenhagen.
Video: link
- Title: The Christoffel function: applications, connections & extensions
Speaker: Jean Bernard Lasserre
Date and Time: Monday, February 17, 2025, at 14h15
Location: Alan Turing Building – Amphithéâtre Sophie Germain
Abstract: We will give a brief introduction to the Christoffel function (CF). One reason and motivation for considering this tool is that, surprisingly, although it has been known for a long time in approximation theory and orthogonal polynomials, it is only recently that its remarkable properties have proven to be very useful in analysis and data mining; for example, for support inference, outlier detection, and density approximation.
But additionally:
– A non-standard use of the CF allows for the interpolation of discontinuous functions without the -Gibbs phenomenon.
– A modification (or regularization) of the CF also allows for the asymptotic approximation of the unknown density of the measure associated with the CF, avoiding the appearance of the equilibrium measure of the support (which is generally unknown).
Finally, we will conclude by revealing some connections of the CF with apparently unrelated fields, e.g., certificates of positivity in real algebraic geometry, the Pell polynomial equation, and the equilibrium measure of a compact.
Video: link
- Title: Introductory talk on Polynomial Optimization and Lasserre hierarchies
Speaker: Igor Klep
Date and time: Monday, February 10, 2025, at 14h15
Location: Alan Turing Building – Amphithéâtre Sophie Germain
Video: link