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arXiv:2305.05123 · Proceedings of the Royal Society of Edinburgh Section A Mathematics

Nonexpansive and noncontractive mappings on the set of quantum pure states

Michiya Mori, Peter Šemrl

Published Jan 17, 2024• 0 citations
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Abstract

Wigner’s theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn’s generalization of Wigner’s theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.

DOI
10.1017/prm.2023.133

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Wigner’s theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn’s generalization of Wigner’s theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.
Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.
Abstract Let HHH be a Hilbert space and P(H)P(H)P(H) be the projective space of all quantum pure states. Wigner’s theorem states that every bijection ϕ ⁣:P(H)→P(H)\phi \colon P(H)\to P(H)ϕ:P(H)→P(H) that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator U ⁣:H→HU\colon H\to HU:H→H. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle π2\frac{\pi }{2}2π​ (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<α≤π40 < \alpha \leq \frac{\pi }{4}0<α≤4π​ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy π4<α<π2\frac{\pi }{4} < \alpha < \frac{\pi }{2}4π​<α<2π​, 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.
The relations between several different forms of Wigner′s theorem are investigated,the descriptions of the theorem in terms of physics and geometry are given.By using operator theory and operator algebra,the equivalence between these different forms of propositions is proved.The results show that if the surjection T:R1(H)→R1(K) preserves the inner product between unit rays,the surjection S:R(H)→R(K) preserves the inner product between rays,the surjection Φ:P1(H)→P1(K) preserves the trace of the product of rank-one projections,and the surjection W:H→K preserves the inner product between vectors,then there exists unit or anti-unitary operators U:H→K,such that Uy∈Tx,r(Uy)=Sx,Φ(Px)=UPx U* and W(x)=φ(x)U(x),in which φ:H→C satisfies |φ(x)|=1.
We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0\varepsilon\geq0ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/29\varepsilon/29ε/2 in the trace metric and 22ε2\sqrt2\varepsilon22​ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε3\varepsilon3ε even under the hypothesis of δδδ-surjectivity, independently of δδδ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0\varepsilon\to0ε→0, but no such modulus can be chosen independently of dimension.
We provide generalizations of the classical Wigner's theorem as well as Uhlhorn's version of Wigner's theorem by considering maps that send rank-$ 1 $ projections to rank-$ n $ projections. Namely, we describe the general form of maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ multiplying $ n $ times the transition probability and maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ sending each complete orthogonal system of rank-$ 1 $ projections to some complete orthogonal system of rank-$ n $ projections.