Stephan WeisMathematicianTrainee Teacher at WaldGymnasium Berlin email: maths@weisstephan.de

• Research in Mathematics.
Geometry of Quantum States: — 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, 86113 (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, 847870 (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, 190192 (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, 317336 (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 maximumentropy 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, 148173 (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, Preimages of extreme points of the numerical range, and applications, Operators and Matrices 10:4, 10431058 (2016). About • arXiv • Journal link We compute preimages of extreme points using Grünbaum's notion of poonem. This allows us to characterize extreme points having a nonunique preimage and to describe closures of sets of 3x3 matrices whose numerical ranges have the same shape. 10) SW, Maximumentropy inference and inverse continuity of the numerical range, Reports on Mathematical Physics 77:2, 251263 (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 maximumentropy inference map for two observables. 9) Leiba Rodman, Ilya M. Spitkovsky, Arleta Szkoła, and SW, Continuity of the maximumentropy 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 semicontinuity of the face function) to study the continuity of maximumentropy states (and of correlations). We explore the case of two observables using preimages of the numerical range. 8) SW, Andreas Knauf, Nihat Ay, and MingJing 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 manyparty quantum correlations and their maximizers. We point out differences to the correlations in probability distributions. 7) SW, Continuity of the maximumentropy inference, Communications in Mathematical Physics 330:3, 12631292 (2014). About • arXiv • MPI MIS Preprint • Journal link We prove an open mapping theorem about the continuity of inference maps for finitelevel quantum states under continuous constraints. We describe the set of maximumentropy inference states under linear constraints. 6) SW, Information topologies on noncommutative state spaces, Journal of Convex Analysis 21:2, 339399 (2014). About • arXiv • MPI MIS Preprint • Journal link The relative entropy defines two topologies on the state space of an Nlevel quantum system. The rItopology extends the Pythagorean and the projection theorem of exponential families to state of nonmaximal 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 maximumentropy inference and a discontinuous entropy distance for 3level quantum systems. 4) SW, Duality of nonexposed faces, Journal of Convex Analysis 19:3, 815835 (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 nonexposed points. 3) SW, A note on touching cones and faces, Journal of Convex Analysis 19:2, 323353 (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, 31683188 (2011). About • arXiv • Journal link (open access) Employing Grünbaum's notion of poonem, we calculate all faces, including the nonexposed faces, for the class of convex sets, which are projections of the state space of an Nlevel quantum system. Correction: Note from Editors, Linear Algebra and its Applications 436 (2012), xixvi: 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 subalgebras employed. The arXivversion has the error removed. 1) Ina Voigt and SW, Polyhedral Voronoi cells, Contributions to Algebra and Geometry 51:2, 587598 (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, 205216. 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, 173180 (2015). About • arXiv • Journal link We summarize geometric ideas about the `boundary' of a Gibbs family of finitelevel quantum states. 2) SW, Discontinuities in the maximumentropy inference, AIP Conference Proceedings 1553, 192199 (2013). About • arXiv • Journal link We argue with the universality of the maximumentropy 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 threelevel quantum system are constructed by excluding characteristics that the state space does not have. Dimension dependent properties of finitelevel 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 higherdimensional 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 8) Choquet’s Theorem for Constrained Sets of Quantum States, talk at the special session "Generalized Numerical Ranges, Operator Theory, and Quantum Information" at the 33rd IWOTA, Krakow, Poland, September 2022, slides (PDF, 564kB) 7) Analysis of Generalized Gibbs States, talk at the conference Entropy 2021: The Scientific Tool of the 21st Century, Porto, Portugal, May 2021, slides (PDF, 1.1MB) 6) Choquet’s Theorem for Constrained Sets of Quantum States, talk at the MIAN online seminar "Quantum Probability, Statistics, Information", Steklov Mathematical Institute, Moscow, Russia, March 2021, video 5) Choquet’s Theorem for Constrained Sets of Quantum States, talk at the Seminari del Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, Spain, June 2021, slides (PDF, 624kB) 4) Geometry of Marginals of Small Quantum Systems, talk at the IX Conference on Quantum Foundations, Córdoba, Argentina, November 2019, slides (PDF, 245kB) 3) Quantum Weak Coin Flipping, talk at the SUMA 2019  Reunión anual de la UMA junto a la SOMACHI, Mendoza, Argentina, September 2019, slides (PDF, 147kB) 2) Classification of joint numerical ranges of three hermitian matrices of size three, talk at The Fourteenth Workshop on Numerical Ranges and Numerical Radii, Munich, Germany, June 2018, slides (PDF, 1.8MB) 1) A new signature of quantum phase transitions from the numerical range, talk at the conference Entropy 2018: From Physics to Information Sciences and Geometry, University of Barcelona, Spain, May 2018, slides (PDF, 268kB) Posters 2) The rIClosure of an Exponential Family and Ground Spaces, poster presented at the Conference "Quantum Information Theory", trimester "Analysis in Quantum Information Theory", Institut Henri Poincaré, Paris, France, December 2017, poster (A0 size, PDF, 863kB) 1) Mysterious Discontinuity of Quantum Correlation, poster presented at the Joint IASICTP School on Quantum Information Processing, Nanyang Executive Centre, Nanyang Technological University, Singapore, January 2016, poster (A2 size, PDF, 535kB) • profiles: arXiv, Google Scholar, LinkedIn, Mathematics Genealogy Project, MathSciNet, ORCiD, ResearchGate, Scopus, Web of Science, XING, 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 → Start here — Spencer Weart's `The Discovery of Global Warming' (AIP) — Hört auf die Wissenschaft!, Gemeinsame Presseinformation von DVGeo, DMV, GDCh und VBio vom 13.01.2020 (in German)  
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