Quantum smooth uncertainty principles for von Neumann bi-algebras

In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify a number of uncertainty principles on quantum symmetries, such as subfactors, fusion bialgebras, etc., studied in quantum Fourier analysis. We also obtain Wigderson–Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.