Generalising Wigner's theorem

TL;DR

This paper extends Wigner's theorem by characterizing linear maps between operator algebras that map rank-$k$ projectors onto rank-$l$ projectors surjectively, providing a complete solution for prime-dimensional Hilbert spaces.

Contribution

It generalizes Wigner's theorem to higher ranks and prime dimensions, offering a full characterization of such linear maps.

Findings

Complete characterization for prime dimensions

Extension of Wigner's theorem to rank-$k$ and rank-$l$ projectors

Surjective maps preserve projector ranks under specified conditions

Abstract

We analyse linear maps of operator algebras $\mathcal{B}_H(\mathcal{H})$ mapping the set of rank-$k$ projectors onto the set of rank-$l$ projectors surjectively. We give a complete characterisation of such maps for prime $n = \dim\mathcal{H}$. The solution is known for $k=l=1$ as the Wigner's theorem.