RepLAB: A Computational/Numerical Approach to Representation Theory

  title={RepLAB: A Computational/Numerical Approach to Representation Theory},
  author={Denis Rosset and Felipe Montealegre-Mora and Jean-Daniel Bancal},
  journal={arXiv: Quantum Physics},
We present a MATLAB/Octave toolbox to decompose finite dimensionial representations of compact groups. Surprisingly, little information about the group and the representation is needed to perform that task. We discuss applications to semidefinite programming. 
Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments
The first example of an extremal point of the quantum set of correlations that admits physically inequivalent quantum realizations is presented, and the results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality.
Certifying Numerical Decompositions of Compact Group Representations
A performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation, as well as a software implementation that can interface with RepLAB.
Mutually unbiased bases: polynomial optimization and symmetry
Sum-of-squares proofs for the (well-known) fact that there do not exist d+ 2 mutually unbiased bases in dimensions d = 2, 3, 4, 5, 6, 7, 8 are obtained.
Noncommutative polynomial optimization under symmetry
We present a general framework to exploit the symmetries present in the Navascués-PironioAćın semidefinite relaxations that approximate invariant noncommutative polynomial optimization problems. We
Universal Approximation Under Constraints is Possible with Transformers
A quantitative constrained universal approximation theorem which guarantees that for any convex or non-convex compact set K and any continuous function f : R → K, there is a probabilistic transformer F̂ whose randomized outputs all lie in K and whose expected output uniformly approximates f .
Bounding and Simulating Contextual Correlations in Quantum Theory
The information cost of simulating contextuality is introduced, which quantifies the additional information required to simulate contextual correlations in either classical or quantum models, and it is shown that the simulation cost can be efficiently bounded using a variant of the hierarchy of semidefinite relaxations.


An Introduction to Random Matrices
Preface 1. Introduction 2. Real and complex Wigner matrices 3. Hermite polynomials, spacings, and limit distributions for the Gaussian ensembles 4. Some generalities 5. Free probability Appendices
SymDPoly: symmetry-adapted moment relaxations for noncommutative polynomial optimization
An algorithm able to discover automatically and exploit the symmetries present in the problem formulation of semidefinite relaxations is presented and the advantages of symmetrization are discussed, namely reductions in memory use, computation time, and increase in the solution precision.
SMILY: A numerical algorithm to decompose unitary representations and compute Clebsch-Gordan coefficients
A numerical algorithm that computes the decomposition of a finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, inspired by notions of quantum
Linear Algebra and its Applications
It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.
Plea for a semidefinite optimization solver in complex numbers – The full report
Four arguments are presented to convince the reader that skipping the transformation phase and using a complex number algorithm, if any, can sometimes have significant benefits and be valid for other problems in which these operations count a great deal in the computing time.
Quantum Inflation: A General Approach to Quantum Causal Compatibility
Causality is a seminal concept in science: any research discipline, from sociology and medicine to physics and chemistry, aims at understanding the causes that could explain the correlations observed
Optimization Methods and Software
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our
Linear representations of finite groups
Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero:
The conformal bootstrap: Theory, numerical techniques, and applications
Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and