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.

中文翻译：

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保持固定量子角的所有量子纯态空间上的映射结构

令$H$ 为希尔伯特空间，$P(H)$ 为所有量子纯态的射影空间。维格纳定理指出，每个保持纯态之间量子角的双射 $\phi \colon P(H)\to P(H)$ 都是由酉算子或反酉算符. Uhlhorn 定理将这一结果推广到双射映射 $\phi $，仅假设在两个方向上保持量子角 $\frac{\pi }{2}$（正交性）。最近，由 Li-Plevnik-Šemrl 和 Gehér 撰写的两篇论文解决了在两个方向上仅保留一个固定量子角 $\alpha $ 的双射的相应结构问题，前提是 $0 < 。\alpha \leq \frac{\pi }{4}$ 成立。在本文中，我们解决了满足 $\frac{\pi }{4} < 的量子角 $\alpha $ 的剩余结构问题。α < \frac{\pi }{2}$，因此完成一个由 Uhlhorn 启动的程序。特别是，事实证明这些映射总是由酉或反酉算子推导出来的，然而，我们的假设比 Wigner 的要弱得多。