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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   $0 &lt; \alpha \leq \frac{\pi }{4}$   holds. In this paper we solve the remaining structural problem for quantum angles   $\alpha $   that satisfy   $\frac{\pi }{4} &lt; \alpha &lt; \frac{\pi }{2}$  , 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.

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