# Symmetry witnesses

## TL;DR

This paper characterizes symmetry witnesses in quantum mechanics, showing that fixed-rank projections and uniform density operators serve as complete symmetry witnesses, extending Wigner's theorem.

## Contribution

It generalizes the concept of symmetry witnesses from pure states to fixed-rank projections and uniform density operators, providing a full classification.

## Key findings

- Fixed-rank projections are symmetry witnesses under mild conditions.
- Uniform density operators of fixed rank are also symmetry witnesses.
- The results extend Wigner's theorem to a broader class of quantum states.

## Abstract

A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner's theorem, the set of pure states, the rank-one projections, is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the set of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e., reasoning in terms of quantum states, the sets of uniform density operators of corresponding fixed rank are symmetry witnesses too.