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

Gehér, G. P. and Mori, M. (2021) The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle. International Mathematics Research Notices. rnab040. ISSN 1687-0247

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To link to this item DOI: 10.1093/imrn/rnab040

## Abstract/Summary

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

Item Type: Article Yes Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 97697 General Mathematics Oxford University Press (OUP)