Brauer Analysis of Graphs and Its Application in the Quantum Entanglement Theory.

Resumen:  Brauer Data Analysis (BDA) is an algebraic tool for analyzing data based on invariants associated with Brauer configuration algebras. The techniques used in this case are similar to those presented in topological data analysis, where invariants associated with vector spaces arising from some combinatorial data are analyzed to capture data features [1].

On the other hand, quantum entanglement theory was introduced in 1935 by Einstein, Podolsky, and Rosen in their famous EPR paper to show the incompleteness of quantum mechanics theory.

EPR’s proposal was untestable for almost thirty years until the introduction of Bell’s inequalities, which revealed that two-particle correlations for the two spin-½ singlet disagree with any local realistic model. Freedman and Clauser realized such an experiment in 1972, followed by Aspect et al. between 1981 and 1982. Meanwhile, the Greenberger-Horne-Zeilinger (GHZ) theorem showed that the concept of EPR’s elements is self-contradictory.

We recall that Clauser, Aspect, and Zeilinger were granted the 2022 Nobel Prize for their experiments regarding quantum entanglement theory [2]. Zeilinger et al. [3–5] recently realized experimental setups based on entanglement by path identity to build Dicke and GHZ quantum entanglement states.

In this talk, we describe a BDA analysis to the graphs obtained by Zeilinger et al. in their experiments. It allows us to interpret such states as suitable Brauer messages.

References
[1] A.M. Cañadas.; I. Gutierrez.; O.M. Mendez. Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory. Symmetry 2024, 16(570).
[2] The Nobel Prize in Physics 2022. NobelPrize.org. Nobel Prize Outreach AB 2024. Available online: https://www.nobelprize.org/prizes/physics/2022/summary/ (accessed on 1 February 2024).
[3] M. Krenn.; X. Gu.; A. Zeilinger. Quantum experiments and graphs I: Multipartite states as coherent superpositions of perfect matchings. Phys. Rev. Lett. 2017, 119, 240403.
[4] X. Gu.; M. Erhard.; A. Zeilinger.; M. Krenn. Quantum experiments and graphs II: Quantum interference, computation, and state generation. Proc. Natl. Acad. Sci. USA 2019, 116, 4147–4155.
[5] Gu, X.; Chen, L.; Zeilinger, A.; Krenn, M. Quantum experiments and graphs. III. High-dimensional and multiparticle entanglement. Phys. Rev. A 2019, 99, 032338.