Quantum 2025 – scientific programme
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MON: Monday Contributed Sessions
MON 19: Foundational / Mathematical Aspects – Quantum Optics and Quantum Information
MON 19.7: Talk
Monday, September 8, 2025, 18:00–18:15, ZHG008
Lie Meets von Neumann for Symmetry Characterisation of Compact Lie Algebras — Emanuel Malvetti1, Robert Zeier2, and •Thomas Schulte-Herbrüggen1 — 1Technical University of Munich (TUM) — 2Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum Control (PGI-8)
Von Neumann’s celebrated double-centraliser theorem completely characterises an operator algebra by its symmetries/commutant. How can this idea be taken over to symmetry-characterise all simple compact Lie algebras (i.e. subalgs of u(N)) in finite dimension N?
Early contributions (inspired by Noether, Artin, van-der-Waerden) see group algebras to (regular representations of) finite groups as first incarnations of von Neumann algebras—still in finite dimensions.
For compact Lie groups and their Lie algebras, we elucidate the add-ons to central isotypic projections (via the commutant to the adjoint representation) that allow for such a full symmetry characterisation. We thus give a general algorithm that identifies a compact simple Lie algebra just from a given set of generators based on its joint symmetries thus substantially driving our earlier work [1-3] to a full classification.
Our algorithmic approach can be applied to problems in various fields such as measurement-based quantum computing, stabiliser design via Clifford algebras, phases of many-body systems—and last but not least quantum control.
[1] J. Math. Phys. 52, 113510 (2011)
[2] Phys. Rev. A 92, 042309 (2015)
[3] J. Math. Phys. 56, 081702 (2015)
Keywords: Symmetries; Lie algebra; von Neumann algebra; Quantum control; Representation theory