# (Co-)type and the linear stability of Wigner's symmetry theorem

## TL;DR

This paper investigates how nearly-symmetrical transformations between finite-dimensional Banach spaces can be approximated by linear maps, with the approximation quality influenced by the spaces' geometric properties.

## Contribution

It establishes a quantitative link between the stability of almost-symmetries and the geometric constants of Banach spaces, extending Wigner's theorem.

## Key findings

- Transformations preserving transition probabilities approximately can be closely approximated by linear maps.
- Approximation quality depends on the type and cotype constants of the Banach spaces.
- Results generalize stability concepts in symmetry transformations to Banach space geometry.

## Abstract

We study the relation between the linear stability of almost-symmetries and the geometry of the Banach spaces on which these transformations are defined. We show that any transformation between finite dimensional Banach spaces that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.