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Wigner's theorem characterizes isometries of the set of all rank oneprojections on a Hilbert space. In metric geometry nonexpansive maps andnoncontractive maps are well studied generalizations of isometries. We showthat under certain conditions Wigner symmetries can be characterized asnonexpansive or noncontractive maps on the set of all projections of rank one.The assumptions required for such characterizations are injectivity orsurjectivity and they differ in the finite and the infinite-dimensional case.Motivated by a recently obtained optimal version of Uhlhorn's generalization ofWigner's theorem, we also give a description of nonexpansive maps which satisfya condition that is much weaker than surjectivity. Such maps do not need to beWigner symmetries. The optimality of all presented results is shown bycounterexamples.

Nonexpansive and noncontractive mappings on the set of quantum pure states