Stephan Weis
MathematicianTrainee Teacher at Wald-Gymnasium Berlin
e-mail: maths@weis-stephan.de
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— linear images of the set of quantum states (numerical ranges and joint numerical ranges)
— quantum marginals
— constrained states in separable Hilbert spaces
— topology of quantum mechanical inference maps
— information geometry and information topology of Gibbs states with several parameters
• Connections to the Sciences. Research in the interdisciplinary field of quantum information theory: correlation, inference, communication, cryptography
• Teaching. I am experienced at teaching exercise classes at the university level, both in the traditional frontal instruction as well as fostering and supervising participatory classes lead by the students. I enjoyed similar participatory teaching at the secondary level in project weeks with students of different ages. I am also trained in digital teaching using video conference tools (BBB and Zoom).
• publications: articles, theses
Preprint
• SW and João Gouveia, Quantum marginals, faces, and coatoms.
arXiv
Refereed journal papers
18) Daniel Plaumann, Rainer Sinn, and SW, Kippenhahn's Theorem for joint numerical ranges and quantum states, SIAM Journal on Applied Algebra and Geometry 5:1, 86-113 (2021).
About • arXiv • Journal link
Kippenhahn's Theorem asserts that the numerical range is the convex hull of an algebraic curve. We generalize the statement to higher dimensions.
17) SW and Maksim E. Shirokov, The face generated by a point, generalized affine constraints, and quantum theory, Journal of Convex Analysis 28:3, 847-870 (2021).
About • arXiv • Journal link
Among others, we show that every state having finite expected values of any two (not necessarily bounded) positive operators admits a decomposition into pure states with the same expected values.
16) SW and Maksim E. Shirokov, Extreme points of the set of quantum states with bounded energy, Russian Mathematical Surveys 76:1, 190-192 (2021).
About • arXiv • Journal link
We show that every extreme point of the set of quantum states with bounded energy is a pure state. We discuss consequences in functional analysis and quantum information theory.
15) SW, A variational principle for ground spaces, Reports on Mathematical Physics 82:3, 317-336 (2018).
About • arXiv • Journal link
We prove a variational formula for ground spaces of a vector space of hermitian matrices. This yields an algebraic characterization of exposed faces of the joint numerical range.
14) Ilya M. Spitkovsky and SW, Signatures of quantum phase transitions from the numerical range, Journal of Mathematical Physics 59:12, 121901 (2018).
About • arXiv • Journal link
We prove connections between the smoothness of the boundary of the numerical range, smoothness of the ground state energy, and continuity of the maximum-entropy inference map.
13) Konrad Szymański, SW, and Karol Życzkowski, Classification of joint numerical ranges of three hermitian matrices of size three, Linear Algebra and its Applications 545, 148-173 (2018).
About • arXiv • Journal link
We characterize the joint numerical range of three 3x3 matrices in terms of flat portions on the boundary (segments and filled ellipses). We discuss examples for the ten possible configurations of flat portions on the boundary.
12) SW, Operator systems and convex sets with many normal cones, Journal of Convex Analysis 25:1, 41–64 (2018).
About • arXiv • Journal link
The state space of an operator system is ubiquitous in quantum mechanics. We study the lattice of its exposed faces, which is isomorphic to the lattice of ground spaces of the operator system.
11) Ilya M. Spitkovsky and SW, Pre-images of extreme points of the numerical range, and applications, Operators and Matrices 10:4, 1043-1058 (2016).
About • arXiv • Journal link
We compute pre-images of extreme points using Grünbaum's notion of poonem. This allows us to characterize extreme points having a non-unique pre-image and to describe closures of sets of 3x3 matrices whose numerical ranges have the same shape.
10) SW, Maximum-entropy inference and inverse continuity of the numerical range, Reports on Mathematical Physics 77:2, 251-263 (2016).
About • arXiv • Journal link
We prove the equivalence of two open mapping statements, for quantum states and pure quantum states respectively. This has applications to the topology of the maximum-entropy inference map for two observables.
9) Leiba Rodman, Ilya M. Spitkovsky, Arleta Szkoła, and SW, Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach, Journal of Mathematical Physics 57:1, 015204 (2016).
About • arXiv • Journal link
We dwell on convex geometry (lower semi-continuity of the face function) to study the continuity of maximum-entropy states (and of correlations). We explore the case of two observables using pre-images of the numerical range.
8) SW, Andreas Knauf, Nihat Ay, and Ming-Jing Zhao, Maximizing the divergence from a hierarchical model of quantum states, Open Systems & Information Dynamics 22:1, 1550006 (2015).
About • arXiv • SFI Working Paper • Journal link
We discuss many-party quantum correlations and their maximizers. We point out differences to the correlations in probability distributions.
7) SW, Continuity of the maximum-entropy inference, Communications in Mathematical Physics 330:3, 1263-1292 (2014).
About • arXiv • MPI MIS Preprint • Journal link
We prove an open mapping theorem about the continuity of inference maps for finite-level quantum states under continuous constraints. We describe the set of maximum-entropy inference states under linear constraints.
6) SW, Information topologies on non-commutative state spaces, Journal of Convex Analysis 21:2, 339-399 (2014).
About • arXiv • MPI MIS Preprint • Journal link
The relative entropy defines two topologies on the state space of an N-level quantum system. The rI-topology extends the Pythagorean and the projection theorem of exponential families to state of non-maximal rank.
5) SW and Andreas Knauf, Entropy distance: New quantum phenomena, Journal of Mathematical Physics 53:10, 102206 (2012).
About • arXiv • MPI MIS Preprint • Journal link
New quantum features are presented: A discontinuous maximum-entropy inference and a discontinuous entropy distance for 3-level quantum systems.
4) SW, Duality of non-exposed faces, Journal of Convex Analysis 19:3, 815-835 (2012).
About • arXiv • MPI MIS Preprint • Journal link
Galois connections relate faces and touching cones of a polar pair of convex bodies. In dimension two this gives a duality (up to multiplicity) between certain singular points and non-exposed points.
3) SW, A note on touching cones and faces, Journal of Convex Analysis 19:2, 323-353 (2012).
About • arXiv • Journal link
We present a systematic analysis of (not necessarily closed) convex sets regarding their lattices of exposed faces and normal cones, and the more general notions of faces and touching cones, respectively.
2) SW, Quantum convex support, Linear Algebra and its Applications 435:12, 3168-3188 (2011).
About • arXiv • Journal link (open access)
Employing Grünbaum's notion of poonem, we calculate all faces, including the non-exposed faces, for the class of convex sets, which are projections of the state space of an N-level quantum system.
Correction: Note from Editors, Linear Algebra and its Applications 436 (2012), xi-xvi: On page xvi, SW points out that in order to obtain formally correct statements, eigenvalues have to be replaced with spectral values in the various sub-algebras employed. The arXiv-version has the error removed.
1) Ina Voigt and SW, Polyhedral Voronoi cells, Contributions to Algebra and Geometry 51:2, 587-598 (2010).
About • arXiv • Journal link (open access)
A Voronoi cell is defined in terms of a point set in Euclidean space. Several cones associated to the point set are used to decide whether the cell is a polyhedron.
Refereed conference papers
4) Atul Singh Arora, Jérémie Roland, and SW, Quantum Weak Coin Flipping, STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, 2019, 205-216.
About • arXiv • Final version (open access) • Journal link
We improve the bias of explicitly known quantum weak coin flipping protocols from 1/6 to 1/10. We present a numerical algorithm to computes the unitaries of protocols having arbitrarily small bias.
3) SW, The MaxEnt extension of a quantum Gibbs family, convex geometry and geodesics, AIP Conference Proceedings 1641, 173-180 (2015).
About • arXiv • Journal link
We summarize geometric ideas about the `boundary' of a Gibbs family of finite-level quantum states.
2) SW, Discontinuities in the maximum-entropy inference, AIP Conference Proceedings 1553, 192-199 (2013).
About • arXiv • Journal link
We argue with the universality of the maximum-entropy inference that the discontinuities of the inference map deserve further investigation. We explain an open mapping condition regarding the continuity of the inference map.
1) Ingemar Bengtsson, SW, and Karol Życzkowski, Geometry of the set of mixed quantum states: An apophatic approach, in: Geometric Methods in Physics, P. Kielanowski, S. T. Ali, A. Odzijewicz, M. Schlichenmaier, T. Voronov (eds.), Basel: Birkhäuser, 2013, 175–197.
About • arXiv • MPI MIS Preprint • Journal link
State space models of a three-level quantum system are constructed by excluding characteristics that the state space does not have. Dimension dependent properties of finite-level quantum systems are revisited.
Academic theses
3) SW, Exponential families with incompatible statistics and their entropy distance, Doctoral Thesis, Erlangen, Germany, 2010.
Electronic Library (open access)
2) SW, Invariants for the ideal boundary of a tree, Diploma Thesis, Erlangen, Germany, 2004.
PDF file (639kB)
1) SW, Dynamics on graphs, Master Thesis, Bristol, UK, 2004.
PDF file (955kB)
Unpublished manuscripts
2) SW, Decomposition of symmetric separable states and ground state energy of bosonic systems.
About • arXiv
We recall that each symmetric separable state admits a convex decomposition into symmetric pure product states. This provides a geometric approach to ground state problems of infinite bosonic systems.
1) SW, On a theorem by Kippenhahn.
About • arXiv
Kippenhahn proved that the numerical range is the convex hull of a real algebraic plane curve. We point out that higher-dimensional analogues are desirable for use in quantum mechanics and they are a challenge in real algebraic geometry.
• presentations: slides, posters, and a video
Slides and a video
6) Analysis of Generalized Gibbs States, Entropy 2021: The Scientific Tool of the 21st Century, Porto, Portugal, May 5th-7th 2021, Slides (PDF, 1.1MB)
5) Choquet’s Theorem for Constrained Sets of Quantum States (with video), MIAN online seminar "Quantum Probability, Statistics, Information", Steklov Mathematical Institute, Moscow, Russia, March 18th 2021; similar talk at the Seminari del Grup d’Informació Quàntica at Universitat Autònoma de Barcelona, Spain, June 3rd 2021, Slides (PDF, 624kB)
4) Geometry of Marginals of Small Quantum Systems, IX Conference on Quantum Foundations, Córdoba, Argentina, November 27th-29th 2019, Slides (PDF, 245kB)
3) Quantum Weak Coin Flipping, SUMA 2019 - Reunión anual de la UMA junto a la SOMACHI, Mendoza, Argentina, September 24th-27th 2019, Slides (PDF, 147kB)
2) Classification of joint numerical ranges of three hermitian matrices of size three, The Fourteenth Workshop on Numerical Ranges and Numerical Radii, Munich, Germany, June 13th-18th 2018, Slides (PDF, 6MB), Slides with less pictures (PDF, 1.8MB)
1) A new signature of quantum phase transitions from the numerical range, Entropy 2018: From Physics to Information Sciences and Geometry, University of Barcelona, Spain, May 14th - 16th 2018, Slides (PDF, 268kB)
Posters
2) The rI-Closure of an Exponential Family and Ground Spaces, Conference "Quantum Information Theory", December 11th-15th 2017, trimester "Analysis in Quantum Information Theory", Institut Henri Poincaré, Paris, France, Poster (A0 size, PDF, 863kB)
1) Mysterious Discontinuity of Quantum Correlation, Joint IAS-ICTP School on Quantum Information Processing, January 18th-29th 2016, Nanyang Executive Centre, Nanyang Technological University, Singapore, Poster (A2 size, PDF, 535kB)
• profiles: arXiv, Google Scholar, LinkedIn, Math-Net.Ru, Mathematics Genealogy Project, MathSciNet, ORCiD, ResearchGate, Scopus, Web of Science, zbMATH
• Erdős number = 3: Chains of this length are
Paul Erdős — Arthur L. Rubin — Leiba Rodman — SW
Paul Erdős — Mario Szegedy — Jérémie Roland — SW
• links regarding climate change
— The RealClimate blog
— Hört auf die Wissenschaft!, Gemeinsame Presseinformation von DVGeo, DMV, GDCh und VBio vom 13.01.2020 (in German)
