# Are almost-symmetries almost linear?

## TL;DR

This paper studies the stability of Wigner's symmetry theorem, showing that near-symmetries in quantum state transitions can be approximated linearly, but such approximations have limitations depending on the Hilbert space dimension.

## Contribution

It establishes bounds on the stability of Wigner's theorem, demonstrating when near-symmetries can be approximated linearly and highlighting the dependence on Hilbert space dimension.

## Key findings

- Transformations approximately preserving transition probabilities admit weak linear approximations.
- A linear approximation close in norm and independent of dimension generally does not exist.
- Lower bounds depend logarithmically on the Hilbert space dimension.

## Abstract

It $d-$pends. Wigner's symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive $\varepsilon$ in a separable Hilbert space admits a weak linear approximation, i.e. one relative to any fixed observable. This implies the existence of a linear approximation that is $4\sqrt{\varepsilon} d$-close in Hilbert-Schmidt norm, with $d$ the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of $d$ does not exist in general. To this end, we provide a lower bound that depends logarithmically on $d$.