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

• Published in: International mathematics research notices Vol. 2022; no. 16; pp. 12003 - 12029
• Main Authors: Gehér, György Pál, Mori, Michiya
• Format: Journal Article
• Language: English
• Published: Oxford University Press 06.08.2022
• Subjects: quantum angle preserving map; Fubini–Study metric; quantum pure state; transition probability preserving map; Primary: 47B49; 51A05; Wigner symmetry; Secondary: 47N50; Projective space
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnab040

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 generalizes 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–Šemrl and Gehér, 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.