Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-02T02:30:07.906Z Has data issue: false hasContentIssue false
  • English
  • Français

New and improved bounds on the contextuality degree of multi-qubit configurations

Published online by Cambridge University Press:  18 April 2024

Axel Muller Open the ORCID record for Axel Muller [Opens in a new window] ,
Metod Saniga Open the ORCID record for Metod Saniga [Opens in a new window] ,
Alain Giorgetti Open the ORCID record for Alain Giorgetti [Opens in a new window] ,
Henri de Boutray  and
Frédéric Holweck
Axel Muller*
Affiliation:
Université de Franche-Comté, CNRS, institut FEMTO-ST, F-25000 Besançon, France
Metod Saniga
Affiliation:
Astronomical Institute of the Slovak Academy of Sciences, 059 60 Tatranska Lomnica, Slovakia
Alain Giorgetti
Affiliation:
Université de Franche-Comté, CNRS, institut FEMTO-ST, F-25000 Besançon, France
Henri de Boutray
Affiliation:
ColibrITD, France
Frédéric Holweck
Affiliation:
ICB, UMR 6303, CNRS, University of Technology of Belfort-Montbéliard, UTBM, 90010 Belfort, France Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA
*
Corresponding author: Axel Muller; Email: axel.muller@femto-st.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

Keywords

quantum geometrymulti-qubit observablesquantum contextualitycontextuality degree
Type
Paper
Information
Mathematical Structures in Computer Science , First View , pp. 1 - 22
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bouillaguet, C. (2017). SpaSM : A Sparse Direct Solver Modulo p, v1.2 edition. http://github.com/cbouilla/spasm.Google Scholar
Budroni, C., Cabello, A., Gühne, O., Kleinmann, M. and Larsson, J.-Å. (2022). Kochen-Specker contextuality. Reviews of Modern Physics 94 045007. https://doi.org/10.1103/RevModPhys.94.045007.CrossRefGoogle Scholar
Cabello, A. (2010). Proposed test of macroscopic quantum contextuality. Physical Review A 82 (3) 032110. https://doi.org/10.1103/PhysRevA.82.032110.CrossRefGoogle Scholar
Chowdhury, M., Müller, M. and You, J. (2021). A deep dive into conflict generating decisions. https://doi.org/10.48550/arXiv.2105.04595.CrossRefGoogle Scholar
De Boeck, M., Rodgers, M., Storme, L. and Švob, A. (2019). Cameron-Liebler sets of generators in finite classical polar spaces. Journal of Combinatorial Theory, Series A 167 340–388. https://doi.org/10.1016/j.jcta.2019.05.005.CrossRefGoogle Scholar
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 55 (47) 475301. https://doi.org/10.1088/1751-8121/aca36f.CrossRefGoogle Scholar
Heawood, P. (1890). Map colouring theorems. The Quarterly Journal of Mathematics, Oxford Series 24 322–339.Google Scholar
Holweck, F. (2021). Testing quantum contextuality of binary symplectic polar spaces on a noisy intermediate scale quantum computer. Quantum Information Processing 20 (7) 247. https://doi.org/10.1007/s11128-021-03188-9.CrossRefGoogle Scholar
Holweck, F., de Boutray, H. and Saniga, M. (2022). Three-qubit-embedded split Cayley hexagon is contextuality sensitive. Scientific Reports 12 (8915). https://doi.org/10.1038/s41598-022-13079-3.CrossRefGoogle ScholarPubMed
Holweck, F. and Saniga, M. (2017). Contextuality with a small number of observables. International Journal of Quantum Information 15 (04) 1750026. https://doi.org/10.1142/S0219749917500265.CrossRefGoogle Scholar
Junttila, T. A. and Niemelä, I. (2000). Towards an efficient tableau method for Boolean circuit satisfiability checking. In Computational Logic – CL 2000, Berlin, Heidelberg. Springer Berlin Heidelberg, 553–567. https://doi.org/10.1007/3-540-44957-4_37.CrossRefGoogle Scholar
Lévay, P. and Szabó, Z. (2017). Mermin pentagrams arising from Veldkamp lines for three qubits. Journal of Physics A: Mathematical and Theoretical 50 (9) 095201. https://doi.org/10.1088/1751-8121/aa56aa.CrossRefGoogle Scholar
Mermin, N. D. (1993). Hidden variables and the two theorems of John Bell. Reviews of Modern Physics 65 803–815. https://doi.org/10.1103/RevModPhys.65.803.CrossRefGoogle Scholar
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 64 101853. https://doi.org/10.1016/j.jocs.2022.101853.CrossRefGoogle Scholar
Peres, A. (1990). Incompatible results of quantum measurements. Physics Letters A 151 (3) 107–108. https://doi.org/10.1016/0375-9601(90)90172-K.CrossRefGoogle Scholar
Planat, M., Saniga, M. and Holweck, F. (2013). Distinguished three-qubit “magicity” via automorphisms of the split Cayley hexagon. Quantum Information Processing 12 (7) 2535–2549. https://doi.org/10.1007/s11128-013-0547-3.CrossRefGoogle Scholar
Polster, B. (1998). Pretty pictures of geometries. Bulletin of the Belgian Mathematical Society – Simon Stevin 5(2/3) 417–425. https://doi.org/10.36045/bbms/1103409021.CrossRefGoogle Scholar
Polster, B., Schroth, A. E., and Van Maldeghem, H. (2001). Generalized flatland. The Mathematical Intelligencer 23 (4) 33–47. https://doi.org/10.1007/BF03024601.CrossRefGoogle Scholar
Saniga, M. (2021). A class of three-qubit contextual configurations located in Fano pentads. Mathematics 9 (13). https://doi.org/10.3390/math9131524.CrossRefGoogle Scholar
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 9 (18) 2272. https://doi.org/10.3390/math9182272.CrossRefGoogle Scholar
Saniga, M. and Lévay, P. (2012). Mermin’s pentagram as an ovoid of PG(3, 2). Europhysics Letters 97 (5) 50006. https://doi.org/10.1209/0295-5075/97/50006.CrossRefGoogle Scholar
Saniga, M., Planat, M., Pracna, P. and Havlicek, H. (2007). The Veldkamp space of two-qubits. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 3 (075) 7. https://doi.org/10.3842/SIGMA.2007.075.Google Scholar