Quantum Geometry

* 2026 *

Lévay, P., and Saniga, M.: 2026, The Finite Geometry of Breaking Quantum Secrets, to be submitted;
[Also https://arXiv.org/abs/2602.08410].

* 2025 *



Muller, A., Saniga, M., Giorgetti, A., Holweck, F., and Kelleher, C.: 2025, A New Heuristic Approach for Contextuality Degree Estimates and its Four- to Six-Qubit Portrayals, Journal of Physics A: Mathematical and Theoretical, Vol. 58, No. 21, 215302 (42 pages).
[Also https://arXiv.org/abs/2407.02928].



Saniga, M., Holweck, F., Kelleher, C., Muller, A., Giorgetti, A., and de Boutray, H.: 2025, Hexagons Govern Three-Qubit Contextuality, Quantum, Vol. 9, 1601 (28 pages).
[Also https://arXiv.org/abs/2312.07738].

* 2024 *



Muller, A., Saniga, M., Giorgetti, A., Holweck, F., and de Boutray, H.: 2024, New and Improved Bounds on the Contextuality Degree of Multi-Qubit Configurations, Mathematical Structures in Computer Science, Vol. 34, Issue 4, pp. 322-343.
[Also https://arXiv.org/abs/2305.10225]. * 2022 *



Muller, A., Saniga, M., Giorgetti, A., de Boutray, H., and Holweck, F.: 2022, Multi-Qubit Doilies: Enumeration for All Ranks and Classification for Ranks Four and Five, Journal of Computational Science, Vol. 64, 101853 (18 pages).
[Also https://arXiv.org/abs/2206.03599].



de Boutray, H., Holweck, F., Giorgetti, A., Masson, P.-A., and Saniga, M.: 2022, Contextuality Degree of Quadrics in Multi-Qubit Symplectic Polar Spaces, Journal of Physics A: Mathematical and Theoretical, Vol. 55, No. 47, 475301 (19 pages).
[Also https://arXiv.org/abs/2105.13798].



Holweck, F., de Boutray, H., and Saniga, M.: 2022, Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive, Scientific Reports, Vol. 12, 8915 (9 pages).
[Also https://arXiv.org/abs/2202.00726].

* 2021 *



Saniga, M., de Boutray, H., Holweck, F., and Giorgetti, A.: 2021, Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank, Mathematics, Vol. 9, No. 18, 2272 (18 pages).
[Also https://arXiv.org/abs/2105.03635].



Saniga, M.: 2021, Taxonomy of Three-Qubit Doilies, an invited minisymposium talk given at the 8th European Congress of Mathematics, Portoroz (Slovenia), on June 23, 2021.



Saniga, M.: 2021, A Class of Three-Qubit Contextual Configurations Located in Fano Pentads, Mathematics, Vol. 9, No. 13, 1524 (6 pages).
[Also http://arXiv.org/abs/2004.07517].



Kelleher, C., Holweck, F., Lévay, P., and Saniga, M.: 2021, X-States From a Finite Geometric Perspective, Results in Physics, Vol. 22C, 103859 (9 pages).
[Also http://arXiv.org/abs/2008.03063].

* 2020 *


Saniga, M., Holweck, F., and Jaffali, H.: 2020, Taxonomy of Three-Qubit Mermin Pentagrams, Symmetry, Vol. 12, No. 4, 534 (7 pages).
[Also http://arXiv.org/abs/1911.11401].

* 2019 *


Saniga, M.: 2019, Doily - A Gem of the Quantum Universe, an invited minisymposium talk given at the 9th Slovenian International Conference on Graph Theory, Bled (Slovenia), on June 28, 2019.

* 2018 *


Saniga, M.: 2018, Finite Geometries Relevant for Quantum Information, an invited talk given at the Graph Theory Seminar, Department of Computer Science, FMPI, Comenius University, Bratislava (Slovakia), on February 22, 2018.



Saniga, M.: 2018, Finite Geometries: Other Notable Examples, an invited talk given at the Algebraic Graph Theory Seminar, Department of Algebra, Geometry and Math Education, FMPI, Comenius University, Bratislava (Slovakia), on February 23, 2018.

* 2017 *


Saniga, M.: 2017, A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line, Entropy, Vol. 19, No. 10, 556 (6 pages). [Also http://arXiv.org/abs/1709.02578].



Saniga, M.: 2017, Polar Spaces and Generalized Polygons Shaping Quantum Information, an invited talk given at the 55-th Summer School on Algebra and Ordered Sets, Nový Smokovec/High Tatras (Slovak Republic), on September 7, 2017.



Saniga, M.: 2017, Polar Spaces and Generalized Polygons with Quantum Physical Flavor, an invited seminar talk given at the Department of Physics and Engineering, Elizabethtown College, Elizabethtown (U. S. A.), on April 24, 2017.



Lévay, P., Holweck, F., and Saniga, M.: 2017, The Magic Three-Qubit Veldkamp Line: A Finite Geometric Underpinning for Form Theories of Gravity and Black Hole Entropy, Physical Review D, Vol. 96, No. 2, 026018 (36 pages).
[Also http://arXiv.org/abs/1704.01598].



Holweck, F., and Saniga, M.: 2017, Contextuality with a Small Number of Observables, International Journal of Quantum Information, Vol. 15, No. 4, 1750026 (12 pages).
[Also http://arXiv.org/abs/1607.07567].



Saniga, M., Holweck, F., and Pracna, P.: 2017, Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups, International Journal of Geometric Methods in Modern Physics, Vol. 14, No. 5, 1750080 (23 pages).
[Also http://hal.archives-ouvertes.fr/hal-01312517 and http://arXiv.org/abs/1605.02001].

* 2015 *


Planat, M., Giorgetti, A., Holweck, F., and Saniga, M.: 2015, Quantum Contextual Finite Geometries from Dessins d'Enfants, International Journal of Geometric Methods in Modern Physics, Vol. 12, No. 7, 1550067 (18 pages).
[Also http://hal.archives-ouvertes.fr/hal-00873461 and http://arXiv.org/abs/1310.4267].

* 2014 *


Holweck, F., Saniga, M., and Lévay, P.: 2014, A Notable Relation Between N-Qubit and 2^{N-1}-qubit Pauli groups via Binary LGr(N, 2N), Symmetry, Integrability and Geometry: Methods and Applications, Vol. 10, Paper 041, 16 pages.
[Also http://hal.archives-ouvertes.fr/hal-00903272, http://arXiv.org/abs/1311.2408, and Oberwolfach Preprint OWP-2013-25 ].

* 2013 *


Saniga, M.: 2013, Finite Geometries with a Quantum Physical Flavor, a series of four introductory lectures delivered at the Laboratory IRTES-M3M, University of Technology of Belfort-Montbeliard, Belfort (France), on September 19 and 26, 2013.



Lévay, P., Planat, M., and Saniga, M.: 2013, Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes, Journal of High Energy Physics, Vol. 09, 037 (35pp).
[Also http://hal.archives-ouvertes.fr/hal-00825701, http://arXiv.org/abs/1305.5689, and Oberwolfach Preprint OWP-2013-17].



Saniga, M.: 2013, Finite Geometries with a Quantum Physical Flavour, an invited talk given at the 14th Conference of Košice's Mathematicians, Herľany (Slovak Republic), April 5, 2013.



Planat, M., Saniga, M., and Holweck, F.: 2013, Distinguished Three-Qubit `Magicity' via Automorphisms of the Split Cayley Hexagon, Quantum Information Processing, Vol. 12, No. 7, pp. 2535-2549.
[Also http://hal.archives-ouvertes.fr/hal-00763975 and http://arXiv.org/abs/1212.2729].

* 2012 *


Saniga, M.: 2012, Finite Ring Geometries of Multi-Qudits and Black Holes, a talk given at my doctoral thesis defense, Comenius University, Bratislava (Slovak Republic), September 25, 2012.



Saniga, M., Planat, M., Pracna, P., and Lévay, P.: 2012, `Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 8, 083 (9pp).
[Also http://hal.archives-ouvertes.fr/hal-00708602 and http://arXiv.org/abs/1206.3436].



Planat, M. and Saniga, M.: 2012, Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry, Physics Letters A, Vol. 376, No. 46, pp. 3485-3490.
[Also http://hal.archives-ouvertes.fr/hal-00703166 and http://arXiv.org/abs/1206.0105].



Saniga, M., and Planat, M.: 2012, Finite Geometry Behind the Harvey-Chryssanthacopoulos Four-Qubit Magic Rectangle, Quantum Information and Computation, Vol. 11, Nos. 11-12, pp. 1011-1016.
[Also http://hal.archives-ouvertes.fr/hal-00692040 and http://arXiv.org/abs/1204.6229].



Saniga, M., Lévay, P., and Pracna, P.: 2012, Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2), Journal of Physics A: Mathematical and Theoretical, Vol. 45, No. 29, 295304 (16pp).
[Also http://hal.archives-ouvertes.fr/hal-00669929 and http://arXiv.org/abs/1202.2973].



Saniga, M.: 2012, Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits, International Journal of Modern Physics B, Vol. 26, Nos. 27-28, 1243013 (3pp).
[Also http://hal.archives-ouvertes.fr/hal-00477098 and http://arXiv.org/abs/1004.4967].



Saniga, M., and Lévay, P.: 2012, Mermin's Pentagram as an Ovoid of PG(3,2), EPL (Europhysics Letters), Vol. 97, No. 5, 50006.
[Also http://hal.archives-ouvertes.fr/hal-00644786 and http://arXiv.org/abs/1111.5923].

* 2011 *


Havlicek, H., Odehnal, B., and Saniga, M.: 2011, Mutually Inscribed and Circumscribed Simplices - Where Moebius Meets Pauli , a seminar talk given at the Department of Mathematics, University of Warmia a Mazury, Olsztyn (Poland), September 21, 2011.



Havlicek, H., Odehnal, B., and Saniga, M.: 2011, On Moebius Pairs of Simplices, a contributed talk given at the 10th conference on Geometry and Applications, held in Varna (Bulgaria), September 3 - 9, 2011.



Havlicek, H., Odehnal, B., and Saniga, M.: 2011, Moebius Pairs and Pauli Operators, a contributed talk given at the conference on Geometry - Theory and Applications, held in Vorau (Austria), June 20 - 24, 2011.



Saniga, M., and Planat, M.: 2011, A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies, Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 22, 225305 (12pp).
[Also http://hal.archives-ouvertes.fr/hal-00566457 and http://arXiv.org/abs/1102.3281].



Planat, M., Lévay, P., and Saniga, M.: 2011, Balanced Tripartite Entanglement, the Alternating Group A_4 and the Lie Algebra sl(3,C) oplus u(1), Reports on Mathematical Physics, Vol. 67, No. 1, pp. 39-51.
[Also http://hal.archives-ouvertes.fr/hal-00437860 and http://arXiv.org/abs/0912.0172].

* 2010 *


Havlicek, H., Odehnal, B., and Saniga, M.: 2010, Moebius Pairs of Simplices and Commuting Pauli Operators, Mathematica Pannonica, Vol. 21, pp. 115-128.
[Also http://hal.archives-ouvertes.fr/hal-00389288 and http://arXiv.org/abs/0905.4648].



Saniga, M., and Pracna, P.: 2010, Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?, Journal of Cosmology (Invited Paper), Vol. 4, pp. 719-735.
[Also http://hal.archives-ouvertes.fr/hal-00308892 and http://arXiv.org/abs/0808.0402].

* 2009 *


Saniga, M.: 2009, From Pauli Groups to Stringy Black Holes: Part II - Generalized Polygons, Geometric Hyperplanes and Some Distinguished Graphs, invited lecture presented within the framework of the ZiF Cooperation Group Finite Projective Ring Geometries, Bielefeld (Germany), August 28, 2009.



Saniga, M.: 2009, From Pauli Groups to Stringy Black Holes: Part I - Projective (Near)Ring Lines, invited lecture presented within the framework of the ZiF Cooperation Group Finite Projective Ring Geometries, Bielefeld (Germany), August 27, 2009.



Saniga, M., Lévay, P., Vrana, P., and Pracna, P.: 2009, GQ(2,4), Split Cayley Hexagon of Order Two and Black Hole Entropy Formulas, an invited talk presented at the miniworkshop on Groups, Discrete Geometry and Quantum Information, held at the Institute of Nuclear Physics of Lyon, Lyon (France), June 11, 2009.



Lévay, P., Saniga, M., Vrana, P., and Pracna, P.: 2009, Black Hole Entropy, Finite Geometry and Mermin Squares, an invited talk presented at Princeton University, Princeton (U. S. A.), May 21, 2009.



Havlicek, H., Odehnal, B., and Saniga, M.: 2009, Factor-Group-Generated Polar Spaces and (Multi-)Qudits, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 5, Paper 096, 15 pages.
[Also http://hal.archives-ouvertes.fr/hal-00372071 and http://arXiv.org/abs/0903.5418].



Lévay, P., Saniga, M., Vrana, P., and Pracna, P.: 2009, Black Hole Entropy and Finite Geometry, Physical Review D, Vol. 79, No. 8, 084036 (12 pages); paper was selected for the May 2009 issue of Virtual Journal of Quantum Information covering a focused area of frontier research.
[Also http://arXiv.org/abs/0903.0541].



Lévay, P., Saniga, M., and Vrana, P.: 2009, Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes, an invited talk presented at the Mathematical Physics Workshop UQ 2009, held at the University of Queensland, Coolangatta (Australia), February 3 - 6, 2009.

* 2008 *


Lévay, P., Saniga, M., and Vrana, P.: 2008, Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes, Physical Review D, Vol. 78, No. 12, 124022 (16 pages); paper was selected for the January 2009 issue of Virtual Journal of Quantum Information covering a focused area of frontier research.
[Also http://hal.archives-ouvertes.fr/hal-00315306 and http://arXiv.org/abs/0808.3849].



Saniga, M., and Pracna, P.: 2008, Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?, presented at "The Clock and the Quantum: Time and Quantum Foundations," held at the Perimeter Institute, Waterloo (Canada), September 28 -- October 2, 2008.
[Also http://hal.archives-ouvertes.fr/hal-00308892 and http://arXiv.org/abs/0808.0402].



Saniga, M., Planat, M., and Pracna, P.: 2008, Projective Ring Line Encompassing Two-Qubits, Theoretical and Mathematical Physics, Vol. 155, No. 3, pp. 905-913.
[Also http://hal.archives-ouvertes.fr/hal-00111733 and http://arXiv.org/abs/quant-ph/0611063].



Havlicek, H., and Saniga, M.: 2008, Qudits and Geometry over Rings, a talk given at the 35th workshop on Geometry and Algebra, held in Berlin (Germany), February 28 - March 4, 2008.



Planat, M., Baboin, A.-C., and Saniga, M.: 2008, Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators, International Journal of Theoretical Physics, Vol. 47, pp. 1127-1135.
[Also http://hal.archives-ouvertes.fr/hal-00147435 and http://arXiv.org/abs/0705.2538].



Havlicek, H., and Saniga, M.: 2008, Projective Ring Line of an Arbitrary Single Qudit, Journal of Physics A: Mathematical and Theoretical, Vol. 41, No. 1, 015302 (12pp).
[Also http://hal.archives-ouvertes.fr/hal-00176551 and http://arXiv.org/abs/0710.0941].



Planat, M., and Saniga, M.: 2008, On the Pauli Graph of N-Qudits, Quantum Information and Computation, Vol. 8, No. 1-2, pp. 0127-0146.
[Also http://hal.archives-ouvertes.fr/hal-00127731 and http://arXiv.org/abs/quant-ph/0701211].


* 2007 *


Saniga, M.: 2007, A Fine Structure of Finite Projective Ring Lines, an invited talk given at the workshop on Prolegomena for Quantum Computing, held in Besançon (France), November 21 -- 22, 2007.
[Also http://hal.archives-ouvertes.fr/hal-00199008].



Havlicek, H., and Saniga, M.: 2007, Projective Ring Line of a Specific Qudit, Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 43, pp. F943-F952.
[Also http://hal.archives-ouvertes.fr/hal-00169103 and http://arXiv.org/abs/0708.4333].



Planat, M., and Saniga, M.: 2007, Finite Geometries and Quantum Information, a talk given at the Colloque Thematique du GdR on Aspects Theoriques de l'Information Quantique, held in Aspet (France), June 7 - 8, 2007.



Saniga, M., Planat, M., Pracna, P., and Havlicek, H.: 2007, The Veldkamp Space of Two-Qubits, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 3, Paper 075, 7 pages.
[Also http://hal.archives-ouvertes.fr/hal-00139548 and http://arXiv.org/abs/0704.0495].



Saniga, M.: 2007, Geometry of Two-Qubits, a seminar talk given at the Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (Czech Republic), on January 25, 2007.



Planat, M., and Saniga, M.: 2007, The N-Qudit Fabric: Pauli Graph and Finite Geometries, an invited talk given at the 10th International Conference on Squeezed States and Uncertainty Relations, held in Bradford (U.K.), March 31 -- April 4, 2007.



Planat, M., and Saniga, M.: 2007, Pauli Graph and Finite Projective Lines/Geometries, a talk given at the biannual SPIE International Congress on Optics and Optoelectronics, held in Prague (Czech Republic), April 16 -- 20, 2007.
Proc. SPIE 6583, 65830W.
[Also http://hal.archives-ouvertes.fr/hal-00137107 and http://arXiv.org/abs/quant-ph/0703154].



Saniga, M., and Planat, M.: 2007, Projective Line over the Finite Quotient Ring GF(2)[x]/(x^3 - x) and Quantum Entanglement: Theoretical Background, Theoretical and Mathematical Physics, Vol. 151, No. 1, pp. 474-481.
[Also http://hal.ccsd.cnrs.fr/ccsd-00020182 and http://arXiv.org/abs/quant-ph/0603051].



Saniga, M., Planat, M., and Minarovjech, M.: 2007, Projective Line over the Finite Quotient Ring GF(2)[x]/(x^3 - x) and Quantum Entanglement: The Mermin "Magic" Square/Pentagram, Theoretical and Mathematical Physics, Vol. 151, No. 2, pp. 625-631.
[Also http://hal.ccsd.cnrs.fr/ccsd-00021604 and http://arXiv.org/abs/quant-ph/0603206].



Saniga, M., and Planat, M.: 2007, Multiple Qubits as Symplectic Polar Spaces of Order Two, Advanced Studies in Theoretical Physics, Vol. 1, No. 1, pp. 1-4.
[Also http://hal.archives-ouvertes.fr/hal-00121565 and http://arXiv.org/abs/quant-ph/0612179].

* 2006 *


Saniga, M.: 2006, Projective Lines over Finite Rings, a seminar talk given at the Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, on September 26, 2006.



Saniga, M.: 2006, Projective Lines over Finite Rings, a seminar talk given at the Research Center for Quantum Information, Institute of Physics, Slovak Academy of Sciences, Bratislava, on August 10, 2006.



Planat, M., Saniga, M., and Kibler, M. R.: 2006, Quantum Entanglement and Projective Ring Geometry, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 2, Paper 066, 14 pages.
[Also http://hal.ccsd.cnrs.fr/ccsd-00077093 and http://arXiv.org/abs/quant-ph/0605239].



Saniga, M., and Planat, M.: 2006, Finite Geometries in Quantum Theory: From Galois (Fields) to Hjelmslev (Rings), International Journal of Modern Physics B, Vol. 20, Nos. 11-13, pp. 1885-1892.



Saniga, M., and Planat, M.: 2006, Hjelmslev Geometry of Mutually Unbiased Bases, Journal of Physics A: Mathematical and General, Vol. 39, No. 2, pp. 435-440.
[Also http://hal.ccsd.cnrs.fr/hal-00005539 and http://arXiv.org/abs/math-ph/0506057].



Saniga, M., and Planat, M.: 2006, Finite Geometries in Quantum Theory: From Galois (Fields) to Hjelmslev (Rings), a poster paper presented at the Ninth Workshop on Quantum Information Processing, held in Paris (France), January 16 - 20, 2006.



* 2005 *


Planat, M., and Saniga, M.: 2005, Galois Algebras of Squeezed Quantum Phase States, Journal of Optics B: Quantum and Semiclassical Optics, Vol. 7, No. 12, pp. S484-S489.
[Also http://hal.ccsd.cnrs.fr/ccsd-00013307].



Planat, M., and Saniga, M.: 2005, Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information, in R. Buccheri, A.C. Elitzur and M. Saniga (eds.), Endophysics, Time, Quantum and the Subjective, World Scientific Publishers, Singapore, pp. 409-426.
[Also http://hal.ccsd.cnrs.fr/ccsd-00004513 and http://arXiv.org/abs/quant-ph/0503159].



Planat, M., and Saniga, M.: 2005, Two-Qutrits from Miniquaternions, a poster paper presented at the EPS-13 Beyond Einstein: Physics for the 21st Century, held in Bern (Switzerland), July 11 - 15, 2005.



Saniga, M., and Planat, M.: 2005, Ovals in Finite Projective Planes and Complete Sets of Mutually Unbiased Bases (MUBs), a poster paper presented at the 9th International Conference on Squeezed States and Uncertainty Relations, held in Besançon (France), May 2 - 6, 2005.



Planat, M., and Saniga, M.: 2005, Galois Algebras of Squeezed Quantum Phase States, a talk given at the 9th International Conference on Squeezed States and Uncertainty Relations, held in Besançon (France), May 2 - 6, 2005.



Saniga, M., and Planat, M.: 2005, Viewing Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes, Chaos, Solitons & Fractals, Vol. 26, No. 5, pp. 1267-1270.
[Also http://hal.ccsd.cnrs.fr/ccsd-00002952 and http://arXiv.org/abs/quant-ph/0409184].


* 2004 *


Planat, M., Rosu, H., Perrine, S. and Saniga, M.: 2004, Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements.
[Also http://arXiv.org/abs/quant-ph/0409081].



Rosu, H., Planat, M., and Saniga, M.: 2004, MUBs: From Finite Projective Geometry to Quantum Phase Enciphering, a poster paper presented at the seventh international conference on Quantum Communication, Measurement and Computing, held in Glasgow (U.K.), July 24-29, 2004; published in CP734, American Institute of Physics, Boston, pp. 315-318.
[Also http://arXiv.org/abs/quant-ph/0409096].



Saniga, M., Planat, M., and Rosu, H.: 2004, Mutually Unbiased Bases and Finite Projective Planes, Journal of Optics B: Quantum and Semiclassical Optics, Vol. 6, No. 9, pp. L19-L20.
[Also http://arXiv.org/abs/math-ph/0403057].