Quantum Fourier analysis

Arthur Jaffe; Chunlan Jiang; Zhengwei Liu; Yunxiang Ren; Jinsong Wu

doi:10.1073/pnas.2002813117

Quantum Fourier analysis

Proc Natl Acad Sci U S A. 2020 May 19;117(20):10715-10720. doi: 10.1073/pnas.2002813117. Epub 2020 Apr 30.

Authors

Arthur Jaffe¹, Chunlan Jiang², Zhengwei Liu^{3

4}, Yunxiang Ren⁵, Jinsong Wu⁶

Affiliations

¹ Harvard University, Cambridge, MA 02138; jaffe@g.harvard.edu liuzhengwei@mail.tsinghua.edu.cn.
² Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050024, China.
³ Yau Mathematical Science Center, Tsinghua University, Beijing 100084, China; jaffe@g.harvard.edu liuzhengwei@mail.tsinghua.edu.cn.
⁴ Department of Mathematics, Tsinghua University, Beijing 100084, China.
⁵ Harvard University, Cambridge, MA 02138.
⁶ Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150027, China.

Abstract

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 [Formula: see text] 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.

Keywords: inequalities; picture language; quantum entanglement; quantum symmetries; uncertainty principles.

Publication types

Research Support, U.S. Gov't, Non-P.H.S.
Research Support, Non-U.S. Gov't