Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T15:18:49.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonexpansive and noncontractive mappings on the set of quantum pure states

Part of: Special classes of linear operators

Published online by Cambridge University Press:  17 January 2024

Michiya Mori Open the ORCID record for Michiya Mori [Opens in a new window]  and
Peter Šemrl Open the ORCID record for Peter Šemrl [Opens in a new window]
Michiya Mori
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan (mmori@ms.u-tokyo.ac.jp) Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Peter Šemrl
Affiliation:
Institute of Mathematics Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia (peter.semrl@fmf.uni-lj.si)
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Wigner symmetrypure statenonexpansive mapnoncontractive map

MSC classification

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 81P05: General and philosophical
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 19
Check for updates
Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burago, D., Burago, Y. and Ivanov, S.. A course in metric geometry, Graduate Studies in Mathematics, Vol. 33 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
Fošner, A., Kuzma, B., Kuzma, T. and Sze, N.-S.. Maps preserving matrix pairs with zero Jordan product. Linear Multi. Algebra 59 (2011), 507–529.CrossRefGoogle Scholar
Gehér, G. P.. An elementary proof for the non-bijective version of Wigner's theorem. Phys. Lett. A 378 (2014), 2054–2057.CrossRefGoogle Scholar
Gehér, G. P. and Mori, M.. The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle. Int. Math. Res. Not. IMRN 2022 (2022), 12003–12029.CrossRefGoogle Scholar
Gehér, G. P. and Šemrl, P.. Isometries of Grassmann spaces, II. Adv. Math. 332 (2018), 287–310.CrossRefGoogle Scholar
Molnár, L.. Generalization of Wigner's unitary-antiunitary theorem for indefinite inner product spaces. Comm. Math. Phys. 210 (2000), 785–791.Google Scholar
Pankov, M.. Wigner-type theorems for Hilbert Grassmannians, London Mathematical Society Lecture Note Series, Vol. 460 (Cambridge University Press, Cambridge, 2020).CrossRefGoogle Scholar
Pankov, M. and Vetterlein, T.. A geometric approach to Wigner-type theorems. Bull. London Math. Soc. 53 (2021), 1653–1662.CrossRefGoogle Scholar
Šemrl, P.. Wigner symmetries and Gleason's theorem. J. Phys. A 54 (2021), 315301, 6pp.CrossRefGoogle Scholar
Uhlhorn, U.. Representation of symmetry transformations in quantum mechanics. Ark. Fysik 23 (1963), 307–340.Google Scholar
Wigner, E. P.. Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum (Frederik Vieweg und Sohn, Braunschweig, 1931).CrossRefGoogle Scholar