Maps on classes of Hilbert space operators preserving measure of commutativity

TL;DR

This paper characterizes transformations on self-adjoint matrices and projections that preserve a unitarily invariant norm of the commutator, extending understanding of measure-preserving maps in quantum operator spaces.

Contribution

It provides a partial answer to a question about norm-preserving maps on self-adjoint matrices and characterizes such maps on rank-one projections in two-dimensional Hilbert spaces.

Findings

Linear transformations preserving commutator norms are described.

Maps on rank-one projections that preserve the quantity are characterized.

Results are applied to describe preservers on the unitary group and density operators.

Abstract

In this paper first we give a partial answer to a question of L. Moln\'ar and W. Timmermann. Namely, we will describe those linear (not necessarily bijective) transformations on the set of self-adjoint matrices which preserve a unitarily invariant norm of the commutator. After that we will characterize those (not necessarily linear or bijective) maps on the set of self-adjoint rank-one projections acting on a two-dimensional complex Hilbert space which leave the latter quantity invariant. Finally, this result will be applied in order to obtain a description of such bijective preservers on the unitary group and on the set of density operators.