Quantum Fourier Analysis

Abstract

{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^p$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outline several open problems.

References in corpus (13)

• Planar algebras, I
• Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration
• Noncommutative Uncertainty Principles
• Planar Para Algebras, Reflection Positivity
• Quons: A 3D Language for Quantum Information
• A noncommutative martingale convexity inequality
• Exchange relation planar algebras of small rank
• Holographic Software for Quantum Networks
• Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
• A Mathematical Picture Language Program
• Uncertainty Principles for Kac Algebras
• Reflection Positivity and Levin-Wen Models
• Non-commutative Rényi Entropic Uncertainty Principles

Cited by in corpus (7)

• Quantum Entropy and Central Limit Theorem
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors
• Interpolated family of non group-like simple integral fusion rings of Lie type
• Quantum smooth uncertainty principles for von Neumann bi-algebras
• Quantum graphs, subfactors and tensor categories I