Quantum Fourier analysis
Abstract
Significance Classical Fourier analysis, discovered over 200 years ago, remains a cornerstone in understanding almost every field of pure mathematics. Its applications in physics range from classical electromagnetism to the formulation of quantum theory. It gives insights into chemistry, engineering, and information science, and it underlies the theory of communication. Quantum Fourier analysis extends this perspective. It yields insights and inequalities associated with uncertainty principles for quantum symmetries. In this paper, we introduce this mathematical subject, we show how it can solve some theoretical problems, and we give some applications to quantum physics with bounds on entropy and the analysis of quantum entanglement. We believe that quantum Fourier analysis, now in its infancy, will have significant future impact. Quantum Fourier analysis is a 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 F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.
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