Research
I work on topics in quantum information theory, mathematical physics, and pure mathematics. I describe some of it below.
Quantum information theory meets von Neumann algebras
I am interested in quantum-information-theoretic properties of systems with unboundedly many degrees of freedom, especially many-body systems in the thermodynamic limit and quantum field theory. I work on these questions together with Alexander Stottmeister, Henrik Wilming, and Reinhard F. Werner. In these settings, local subsystems are naturally described by von Neumann algebras acting on the Hilbert space of the sector under consideration.
A key idea of this line of work is that algebraic properties of these algebras encode operational properties of the subsystems, while the relative position of algebras reflects the quantum correlations (entanglement) between the subsystems. I like to call this the algebraic-operational correspondence. A boring example is that a subsystem is “purely quantum” (no information can be extracted without disturbing the subsystem’s state) precisely when its algebra is a factor (only scalars commute with all other elements of the algebra). More interestingly, we showed that the type classification of factors (types I, II, III and their respective subtypes) is captured by a family of operational entanglement properties of the bipartite system of a factor and its commutant. For instance, type I is characterized by having finite one-shot entanglement, while the Connes classification of type III factors is captured by the system’s performance at the task of “embezzling” entanglement.
I also like to study the operator-algebraic properties in concrete physical models. Combined with general results relating algebraic and operational properties, this reveals new information-theoretic features of those models. This is exactly the strategy we used for critical fermion chains: after identifying their half-chain algebras as type III\(_1\) factors in Haag duality, we could conclude that all states in the ground-state sector can be used to embezzle entanglement. This work was published in Nature Physics.
Publications
Uniqueness of purifications is equivalent to Haag duality
Lauritz van Luijk, Alexander Stottmeister, and Henrik Wilming.
Physical Review Letters 136, 040203 (2026). DOI · arXiv
Pure state entanglement and von Neumann algebras
Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming.
Communications in Mathematical Physics 406, 296 (2025). DOI · arXiv
More publications
Quantum steering is equivalent to state-preserving conditional expectations
Lauritz van Luijk, Amine Marrakchi, Tobias Osborne, Alexander Stottmeister, and Henrik Wilming.
Preprint, 2026. arXiv
Critical fermions are universal embezzlers
Lauritz van Luijk, Alexander Stottmeister, and Henrik Wilming.
Nature Physics 21, 1141–1146 (2025). DOI · arXiv
Multipartite embezzlement of entanglement
Lauritz van Luijk, Alexander Stottmeister, and Henrik Wilming.
Quantum 9, 1818 (2025). DOI · arXiv
Relativistic quantum fields are universal entanglement embezzlers
Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming.
Physical Review Letters 133, 261602 (2024). DOI · arXiv
Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras
Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming.
Preprint, 2024. arXiv
Rényi divergences, Petz recovery and sufficiency with positive trace-preserving maps
In this paper, Niklas Galke, Henrik Wilming and I conjectured that a pair of quantum states can be interconverted via PTP maps with another pair if and only if sufficiently many quantum divergences are equal. More concretely, we conjectured this for the sandwiched quantum Rényi divergence. Originally, the question came about for CPTP maps, but we soon realized that a generic pair of states and its transpose cannot be interconverted by CPTP maps. The transpose is, however, not seen by any of the quantum divergences studied in the literature (although invariance under the transpose is usually not an axiom). While we proved the conjecture in some subcases, the general case remains wide open.
Motivated by this, Henrik and I set out to generalize the Petz recovery theorem (quantum relative entropy + alpha-z quantum Rényi divergences) to the class of PTP maps. In ongoing work with Anna Jenćova and a jointly supervised master’s student, we’re taking this to the level of von Neumann algebras.
Publications
State- and energy-dependent convergence rates
Another line of my work concerns dynamics and approximation of dynamics in finite and infinite-dimensional quantum systems. In various projects, I worked on proving explicit state-dependent convergence rates for the Trotter product formula and related dynamical approximations. I showed how to obtain upper bounds on the energy-increase of dynamics, and how this can be combined with a novel submultiplicativity inequality for the energy-constrained operator or diamond norms. In collaboration with Simon Becker, Niklas Galke, and Robert Salzmann, we applied these and other techniques to establish convergence rates of the Trotter product formula for simulating various Schrödinger and Dirac operators with singular, magnetic, or confining potentials.
Publications
Energy-limited quantum dynamics
Lauritz van Luijk.
Communications in Mathematical Physics 406, 120 (2025). DOI · arXiv
Convergence rates for the Trotter splitting for unbounded operators
Simon Becker, Niklas Galke, Lauritz van Luijk, and Robert Salzmann.
Foundations of Computational Mathematics (2025). DOI · arXiv
More publications
Error bounds for Lie group representations in quantum mechanics
Lauritz van Luijk, Niklas Galke, Alexander Hahn, and Daniel Burgarth.
Journal of Physics A: Mathematical and Theoretical 57, 105301 (2024). DOI · arXiv
Daniel Burgarth, Niklas Galke, Alexander Hahn, and Lauritz van Luijk.
Physical Review A 107, L040201 (2023). DOI · arXiv
Operator algebras and operator theory
I like to work on projects in pure mathematics from time to time, often inspired by the mathematics used to describe quantum theory. Recently, I worked on a projection on soft inductive limits of operator systems and C*-algebras with Kristin Courtney, Niklas Galke, and Alexander Stottmeister. We use a generalization of inductive limits of Banach spaces, introduced in this paper by Alexander Stottmeister, Reinhard F. Werner, and myself, to prove a non-commutative version of the Lazar-Lindenstrass theorem, a classical theorem in convex geometry stating that every Choquet simplex is the projective limit of finite-dimensional simplices.
Publications
Soft inductive limits of operator systems and a noncommutative Lazar–Lindenstrauss theorem
Kristin Courtney, Niklas Galke, Lauritz van Luijk, and Alexander Stottmeister.
Preprint, 2025. To appear in: Journal of Operator Theory. arXiv
Heisenberg-smooth operators from the phase-space perspective
Robert Fulsche and Lauritz van Luijk
Mathematische Nachrichten 298, 2845–2866 (2025). DOI · arXiv
More publications
A simple criterion for essential self-adjointness of Weyl pseudodifferential operators
Robert Fulsche and Lauritz van Luijk.
Journal of Pseudo-Differential Operators and Applications 16, 38 (2025). DOI · arXiv
Self-adjointness of Toeplitz operators on the Segal–Bargmann space
Wolfram Bauer, Lauritz van Luijk, Alexander Stottmeister, and Reinhard F. Werner.
Journal of Functional Analysis 284, 109778 (2023). DOI · arXiv
Some of my works do not fit into the topics above. A complete list of my works is available on the Publications page.