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 $\phi \colon P(H)\to P(H)$ 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.

(No) Wigner Theorem for C*-algebras

Wigner’s Theorem states that bijections of the set [Formula: see text] of one-dimensional projections on a Hilbert space [Formula: see text] that preserve transition probabilities are induced by either a unitary or an anti-unitary operator on [Formula: see text] (which is uniquely determined up to a phase). Since elements of [Formula: see text] define pure states on the C*-algebra [Formula: see text] of all bounded operators on [Formula: see text] (though typically not producing all of them), this suggests possible generalizations to arbitrary C*-algebras. This paper is a detailed study of this problem, based on earlier results by R. V. Kadison (1965), F. W. Shultz (1982), K. Thomsen (1982), and others. Perhaps surprisingly, the sharpest known version of Wigner’s Theorem for C*-algebras (which is a variation on a result from Shultz, with considerably simplified proof) generalizes the equivalence between the hypotheses in the original theorem and those in an analogous result on (anti-)unitary implementability of Jordan automorphisms of [Formula: see text], and does not yield (anti-)unitary implementability itself, far from it: abstract existence results that do give such implementability seem arbitrary and practically useless. As such, it would be fair to say that there is no Wigner Theorem for C*-algebras.

Wigner's theorem for an infinite set

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space ${\mathcal H}$ correspond to direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.