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Wigner’s Theorem and geometry of extreme positive maps

Janusz Grabowski, Cardinal Stefan

Published Jan 1, 2009• 5 citations
OpenAlex

Abstract

We consider transformation maps on the space of states which are symmetries in the sense of Wigner. Due to the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body, to be able to contribute to the classification of positive maps. Our study provides a variant of Wigner’s theorem originally proved for ray transformations in Hilbert spaces.

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We consider transformation maps on the space of states which are symmetries in the sense of Wigner. Due to the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body, to be able to contribute to the classification of positive maps. Our study provides a variant of Wigner’s theorem originally proved for ray transformations in Hilbert spaces.
Providing an operational characterization of the Wigner-positive states (WPS), i.e., the set of quantum states with non-negative Wigner function, is a longstanding open problem. For pure states, the only WPS are Gaussian states, but the situation is considerably more subtle for mixed states. Here, we approach the problem using convex geometry, reducing the question to the characterization of the extreme points of the set of WPS. We give a constructive method to generate a large class of such extreme WPS, which combines the following steps: (i) we characterize the phase-invariant extreme points of the superset of Wigner-positive quasi-states (WPQS); (ii) we introduce a new quantum map, named Vertigo map, which maps extreme WPQS to extreme WPS while preserving phase invariance; (iii) we identify families of extremality-preserving maps and use them to obtain extreme WPS while relaxing phase invariance. Our construction generates all extreme WPS of low dimension, starting from a specific kind of WPS known as beam-splitter states. Our results build upon new mathematical properties of the set of WPS derived in a companion paper and unveil the remarkable structure of mixed states with non-negative Wigner functions.
For Hilbert spaces H⊆L2(R)\mathcal H\subseteq L^2(\mathbb R)H⊆L2(R) we consider the convex sets D+(H)\mathcal D_+(\mathcal H)D+​(H) of Wigner-positive states (WPS), i.e.~density matrices over H\mathcal HH with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate D+(H)\mathcal D_+(\mathcal H)D+​(H) through convex combinations. If H\mathcal HH is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case H=L2(R)\mathcal H=L^2(\mathbb R)H=L2(R) this is not so, due to lack of compactness of the set D+(H)\mathcal D_+(\mathcal H)D+​(H). Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.
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.
Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let H1\mathcal{H}_1H1​ and H2\mathcal{H}_2H2​ be Hilbert spaces, and let kkk be a positive integer such that 2≤k<12dim⁡(H1)2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)2≤k<21​dim(H1​). We show that if an injective map ψ:Pk(H1)→Pk(H2)\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)ψ:Pk​(H1​)→Pk​(H2​) satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let M\mathcal{M}M be a type II1_11​ or type II∞_\infty∞​ factor, let τM\tau_\mathcal{M}τM​ be a faithful normal semi-finite tracial weight of M\mathcal MM, and let $0
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.