The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

TL;DR

This paper characterizes the structure of maps on quantum pure states that preserve a specific quantum angle, extending classical theorems to a broader range of angles and showing such maps are induced by unitary or antiunitary operators.

Contribution

It completes the classification of angle-preserving maps for all angles between  and , generalizing Uhlhorn's theorem and previous results.

Findings

Maps preserving angles  <  are induced by unitary or antiunitary operators.

The result applies to a broader class of angle-preserving maps than previously known.

The assumption on the maps is weaker than the conditions in Wigner's theorem.

Abstract

Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner's theorem states that every bijection $\phi\colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn's theorem generalises this result for bijective maps $\phi$ that are only assumed to preserve the quantum angle $\frac{\pi}{2}$ (orthogonality) in both directions. Recently, two papers, written by Li--Plevnik--\v{S}emrl and Geh\'er, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha$ in both directions, provided that $0 < \alpha \leq \frac{\pi}{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha$ that satisfy $\frac{\pi}{4} < \alpha < \frac{\pi}{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner's.