(with Nikodym and Banach in Cracow) ## Stephan WeisPostdoctoral ScholarContactCentre for Quantum Information and Communication Ecole Polytechnique de Bruxelles Université libre de Bruxelles 50 av. F.D. Roosevelt - CP165/59 B-1050 Bruxelles Belgium phone: +32-2-650 29 72 e-mail: maths@weis-stephan.de |
• research interestsnumerical range lattices of ground spaces maximum-entropy states quantum correlations weak coin flipping • public internet profiles
• arXiv • Google Scholar • ResearchGate • ResearcherID (Web of Science™) • ORCID • currículo lattes (CNPq) (in Portuguese) • detailed list of publications
Preprints• S. Weis (2017) A variation principle for ground spacesAbout • arXiv A variation formula is proved for ground spaces of a vector space of energy operators. The formula is derived from the geometry of normal cones of a state space and is demonstrated with two-local three-bit Hamiltonians. Articles submitted to research journals• I. M. Spitkovsky, S. Weis (submitted) A new signature of quantum phase transitions from the numerical rangeAbout • arXiv Non-analytic points of class C ^{2} on the boundary of the numerical range are signatures of quantum phase transitions. We support this suggestion by
showing that a maximum-entropy inference map is discontinuous at these points.
• K. Szymański, S. Weis, K. Życzkowski (submitted) Classification of joint numerical ranges of three hermitian matrices of size threeAbout • arXiv The joint numerical range of three 3x3 matrices is characterized in terms of its flat boundary portions (segments and filled ellipses). Examples are given for the ten possible three-dimensional objects. Articles accepted in research journals• S. Weis (in print) Operator systems and convex sets with many normal cones,Journal of Convex Analysis 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. Articles published in research journals11) I. M. Spitkovsky, S. Weis (2016) Pre-images of extreme points of the numerical range, and applications,Operators and Matrices 10(4) 1043-1058 About • arXiv • Journal link Grünbaum's notion of poonem allows to compute pre-images of extreme points. A non-unique pre-image is then characterizable in terms of Hausdorff convergence of facets. An application is to describe closures of sets of 3x3 matrices whose numerical ranges have the same shape. 10) S. Weis (2016) Maximum-entropy inference and inverse continuity of the numerical range,Reports on Mathematical Physics 77(2) 251-263 About • arXiv • Journal link Continuity of the maximum-entropy inference which refers to two quantum observables is proven equivalent to the inverse continuity of numerical range points. This yields a continuity condition depending on analytic eigenfunctions. 9) L. Rodman, I. M. Spitkovsky, A. Szkoła, S. Weis (2016) Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach,Journal of Mathematical Physics 57(1) 015204 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) S. Weis, A. Knauf, N. Ay, M.-J. Zhao (2015) Maximizing the divergence from a hierarchical model of quantum states,Open Systems & Information Dynamics 22(1) 1550006 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) S. Weis (2014) Continuity of the maximum-entropy inference,Communications in Mathematical Physics 330(3) 1263-1292 About • arXiv • MPI MIS Preprint • Journal link A continuity criterion is proven for the inference of the state of a finite-level quantum system under linear constraints. The set of inference states of the maximum-entropy inference is described. 6) S. Weis (2014) Information topologies on non-commutative state spaces,Journal of Convex Analysis 21(2) 339-399 About • arXiv • MPI MIS Preprint • Journal link The Umegaki relative entropy defines topologies on the state space of an N-level quantum system. The rI-topology extends Pythagorean and projection theorems for exponential families. 5) S. Weis, A. Knauf (2012) Entropy distance: New quantum phenomena,Journal of Mathematical Physics 53(10) 102206 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) S. Weis (2012) Duality of non-exposed faces,Journal of Convex Analysis 19(3) 815-835 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) S. Weis (2012) A note on touching cones and faces,Journal of Convex Analysis 19(2) 323-353 About • arXiv • Journal link Touching cones of a convex set, that is non-empty faces of normal cones, are explored. 2) S. Weis (2011) Quantum convex support,Linear Algebra and its Applications 435(12) 3168-3188 About • arXiv • Journal link All faces, including the non-exposed faces, are calculated algebraically for projections of the state space of an N-level quantum system. The projections are convex duals of sections of the state space.
2a) S. Weis (2012)
Correction, ibid. 436(1) xviAbout • Journal link Spectral values have to be used in place of eigenvalues. The arXiv-version has this error removed, see also Remark 6 in S. Weis, A. Knauf (2012) Entropy distance: New quantum phenomena,
JMP 53(10) 102206
1) I. Voigt, S. Weis (2010) Polyhedral Voronoi cells,Contributions to Algebra and Geometry 51(2) 587-598 About • arXiv • Journal link 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 if the cell is a polyhedron. Articles published in peer-reviewed conference proceedings3) S. Weis (2015) The MaxEnt extension of a quantum Gibbs family, convex geometry
and geodesics,
34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering,
Château du Clos Lucé, Amboise, France, 21-26 September 2014,
eds. A. Mohammad-Djafari, F. Barbaresco,
AIP Conference Proceedings 1641 173-180About • arXiv • Journal link We summarize geometric ideas about the `boundary' of a Gibbs family of finite-level quantum states. The `boundary' consists of ultra cold quantum states. We prove a zero-temperature representation of the irreducible correlation. 2) S. Weis (2013) Discontinuities in the maximum-entropy inference,
32nd International Workshop on Bayesian Inference and Maximum Entropy Methods
in Science and Engineering, Garching, Germany, 15-20 July 2012, ed. U. von Toussaint,
AIP Conference Proceedings 1553 192-199About • arXiv • Journal link We argue with the universality of the maximum-entropy inference that discontinuities of the maximum-entropy inference deserve further investigation. We explain an openness condition for the continuity of the inference. 1) I. Bengtsson, S. Weis, K. Życzkowski (2013) Geometry of the set of mixed quantum states: An apophatic approach,
Geometric Methods in Physics, XXX Workshop, Białowieża, Poland,
June 26 to July 2, 2011, eds. P. Kielanowski, S. T. Ali, A. Odzijewicz, M. Schlichenmaier,
T. Voronov, Springer Basel, Trends in Mathematics 175-197About • 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 theses3) S. Weis (2010) Exponential families with incompatible statistics and their entropy distance,
Doctoral Thesis, University of Erlangen, GermanyElectronic Library 2) S. Weis (2004) Invariants for the ideal boundary of a tree,
Diploma Thesis, University of Erlangen, GermanyPDF file (639kB) 1) S. Weis (2004) Dynamics on graphs,
Master Thesis, University of Bristol, Bristol, UKPDF file (955kB) Unpublished manuscripts• S. Weis (2017) On a theorem by KippenhahnAbout • 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 and travelling
Upcoming events
• Conference talk: A variation principle for ground spaces,
Workskop on Operator Theory, Complex Analysis, and Applications,
July 3-6 2017,
Department of Mathematics of Instituto Superior Técnico,
University of Lisbon,
Portugal
• Seminar talk: A new signature of quantum phase transitions from the numerical range,
July 10 2017,
Departamento de Matemática,
Algebra and Combinatorics group,
University of Coimbra,
Portugal
• Conference talk: A new signature of quantum phase transitions from the numerical range,
Mini-Symposium on Quantum Information Science,
2017 Meeting of the International Linear Algebra Society,
July 24-28 2017, Department of Mathematics at Iowa State University,
USA
• Sums of Squares - Real Algebraic Geometry and its Applications, 21-25 August 2017, University of Innsbruck, Austria • Trimester Analysis in Quantum Information Theory, 4 September - 15 December 2017, Institut Henri Poincaré, Paris, France Slides from conference talks
• Conference talk: On Lattices of Ground Spaces,
49 Symposium on Mathematical Physics,
June 17-18 2017,
Institute of Physics,
Nicolaus Copernicus University,
Toruń,
Poland,
Slides (PDF, 990 kB)
• A variation principle for ground spaces,
8th Linear Algebra Workshop,
June 12-16 2017,
Faculty of Mathematics and Physics,
Ljubljana,
Slovenia,
Slides (PDF, 483 kB)
• Continuity of many-party correlations,
Invited Minisymposium Linear Algebra and Quantum Computation,
20th Conference of the International Linear Algebra Society,
July 11-15, 2016,
KU Leuven,
Leuven,
Belgium,
Slides (PDF, 578kB)
• A classification of the joint numerical range of three hermitian 3-by-3 matrices,
Workshop on Positive Semidefinite Rank, Program
Semidefinite and Matrix Methods for Optimization and Communication,
February 1-21, 2016,
Institute for Mathematical Sciences,
National University of Singapore, Singapore,
Slides (PDF, 1.1 MB)
Posters presented at conferences
• Information Topologies on Non-Commutative State Spaces,
20th Annual Conference on
Quantum Information Processing,
January 14–20, 2017, Seattle, Washington,
Poster (A0 size, PDF, 880kB)
• Mysterious Discontinuity of Quantum Correlation,
Joint IAS-ICTP School on Quantum Information Processing,
January 18-29, 2016,
Nanyang Executive Centre,
Nanyang Technological University, Singapore,
Poster (A2 size, PDF, 535kB)
• Computing many-party quantum correlations — analytical results, Conference
Theory of Quantum Computation,
Communication and Cryptography,
May 20–22, 2015, Brussels, Belgium,
Poster (A0 size, PDF, 2.1MB)
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