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 $P(H)$ 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

Isometric study of Wasserstein spaces - the real line

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal {W}_2(\mathbb{R}^n)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\mathrm {Isom}(\mathcal {W}_p(\mathbb{R}))$, the isometry group of the Wasserstein space $\mathcal {W}_p(\mathbb{R})$ for all $p \in [1, \infty )\setminus \{2\}$. We show that $\mathcal {W}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $\mathcal {W}_p(\mathbb{R})$ is isometrically rigid if and only if $p\neq 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $\mathcal {W}_p([0,1])$ is isometrically rigid if and only if $p\neq 1$. Moreover, $\mathcal {W}_1([0,1])$ admits isometries that split mass, and $\mathrm {Isom}(\mathcal {W}_1([0,1]))$ cannot be embedded into $\mathrm {Isom}(\mathcal {W}_1(\mathbb{R}))$