Noncommutative polynomial optimization under symmetry

• URL: http://arxiv.org/abs/2112.10803v2
• Date: Thu, 30 Jun 2022 14:26:55 GMT
• Title: Noncommutative polynomial optimization under symmetry
• Authors: Marie Ioannou and Denis Rosset
• Abstract summary: We present a general framework to exploit the symmetries present in the Navascu'es-Pironio-Ac'in semidefinite relaxations. We put equal emphasis on the moment and sum-of-squares dual approaches.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a general framework to exploit the symmetries present in the Navascu{\'e}s-Pironio-Ac{\'i}n semidefinite relaxations that approximate invariant noncommutative polynomial optimization problems. We put equal emphasis on the moment and sum-of-squares dual approaches, and provide a pedagogical and formal introduction to the Navascu{\'e}s-Pironio-Ac{\'i}n technique before working out the impact of symmetries present in the problem. Using our formalism, we compute analytical sum-of-square certificates for various Bell inequalities, and prove a long-standing conjecture about the exact maximal quantum violation of the CGLMP inequalities for dimension 3 and 4. We also apply our technique to the Sliwa inequalities in the Bell scenario with three parties with binary measurements settings/outcomes. Symmetry reduction is key to scale the applications of the NPA relaxation, and our formalism encompasses and generalizes the approaches found in the literature.

Related papers

• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• A field theory representation of sum of powers of principal minors and physical applications [0.0]
  We introduce a novel field theory representation for the Sum of Powers of Principal Minors (SPPM) It offers deeper insights into the symmetries of complex quantum systems. This work bridges theoretical gaps in understanding principal minors within quantum systems.
  arXiv  Detail & Related papers  (2024-03-14T21:09:46Z)
• First-principles construction of symmetry-informed quantum metrologies [0.0]
  We develop a class of measurement strategies for quantities isomorphic to location parameters. The resulting framework admits any parameter range, prior information, or state. It reduces the search for good strategies to identifying which symmetry leaves a state of maximum ignorance invariant.
  arXiv  Detail & Related papers  (2024-02-26T09:06:37Z)
• Multiplicative Updates for Online Convex Optimization over Symmetric Cones [28.815822236291392]
  We introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer.
  arXiv  Detail & Related papers  (2023-07-06T17:06:43Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities [91.46841922915418]
  We present a unified approach for the theoretical analysis of first-order variation methods. Our approach covers both non-linear gradient and strongly Monte Carlo problems. We provide bounds that match the oracle strongly in the case of convex method optimization problems.
  arXiv  Detail & Related papers  (2023-05-25T11:11:31Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Optimal variance-reduced stochastic approximation in Banach spaces [114.8734960258221]
  We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. We establish non-asymptotic bounds for both the operator defect and the estimation error.
  arXiv  Detail & Related papers  (2022-01-21T02:46:57Z)
• Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory [26.555110725656963]
  We show examples of Lieant's theorem, Lie groups, Lie algebras, and the Harish-Chandra--Itzyintegrals formulas. We then present an introduction to optimization theory -- an indispensable mathematical toolkit for capturing continuous symmetries.
  arXiv  Detail & Related papers  (2021-09-02T16:44:44Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing [0.0]
  We present a method to solve the quantum marginal problem for symmetric $d$-level systems. We illustrate the applicability of the method in central quantum information problems with several exemplary case studies.
  arXiv  Detail & Related papers  (2020-01-13T18:20:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.