Quantum Fourier analysis

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...

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Published inProceedings of the National Academy of Sciences - PNAS Vol. 117; no. 20; pp. 10715 - 10720
Main Authors Jaffe, Arthur, Jiang, Chunlan, Liu, Zhengwei, Ren, Yunxiang, Wu, Jinsong
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences 19.05.2020
Subjects
Entropy
Fourier analysis
Fourier transforms
Physical Sciences
Quantum phenomena
quantum entanglement
quantum symmetries
inequalities
uncertainty principles
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Summary: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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Reviewers: L.G., University of New Hampshire and Academy of Mathematics and Systems Science, Chinese Academy of Sciences; and J.W.H., University of California at San Diego.
1A.J., C.J., Z.L., Y.R., and J.W. contributed equally to this work.
Contributed by Arthur Jaffe, February 20, 2020 (sent for review February 14, 2020; reviewed by Liming Ge and J. William Helton)
Author contributions: A.J., C.J., Z.L., Y.R., and J.W. designed research; A.J., C.J., Z.L., Y.R., and J.W. performed research; and A.J., Z.L., Y.R., and J.W. wrote the paper.
ISSN:0027-8424
1091-6490
1091-6490
DOI:10.1073/pnas.2002813117