Quantum Permutation Matrices

Abstract

Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.

Dedicated to Jörg Eschmeier who sadly passed away in 2021.

Communicated by Mihai Putinar.

This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

This work has been supported by the SFB-TRR 195, by the Heisenberg program of the DFG (German Research Foundation) and by OPUS LAP: quantum groups, graphs and symmetries via representation theory jointly funded by DFG and NCN (Poland).