Paper detail

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

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 $ϕ\colon P(H)\to P(H)$ 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 $ϕ$ that are only assumed to preserve the quantum angle $\fracπ{2}$ (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 $α$ in both directions, provided that $0 < α\leq \fracπ{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $α$ that satisfy $\fracπ{4} < α< \fracπ{2}$, 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.