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

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