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An approach using geometric diagrams to generic Bell inequalities with multiple observables

Junghee Ryu, Jinhyoung Lee, Hoon Ryu

Abstract

We extend the generic Bell inequalities suggested by Son, Lee, and Kim [Phys. Rev. Lett. 96, 060406 (2006)] to incorporate multiple observables for tripartite systems and introduce a geometric methodology for calculating classical upper bounds of the inequalities. Our method transforms the problem of finding the classical upper bounds into identifying constraints in linear congruence relations. Using this approach, we derive the upper bounds for scenarios with three and four observables per party. In order to demonstrate quantum violations, we employ Greenberger-Horne-Zeilinger entangled states that can achieve values exceeding the classical upper bounds, with the violation becoming more pronounced as the number of observables increases.

An approach using geometric diagrams to generic Bell inequalities with multiple observables

Abstract

We extend the generic Bell inequalities suggested by Son, Lee, and Kim [Phys. Rev. Lett. 96, 060406 (2006)] to incorporate multiple observables for tripartite systems and introduce a geometric methodology for calculating classical upper bounds of the inequalities. Our method transforms the problem of finding the classical upper bounds into identifying constraints in linear congruence relations. Using this approach, we derive the upper bounds for scenarios with three and four observables per party. In order to demonstrate quantum violations, we employ Greenberger-Horne-Zeilinger entangled states that can achieve values exceeding the classical upper bounds, with the violation becoming more pronounced as the number of observables increases.

Paper Structure

This paper contains 9 sections, 1 theorem, 33 equations, 3 figures.

Table of Contents

  1. Generic Bell operator with multiple observables
  2. Results and discussion
  3. Conclusion
  4. Data availability
  5. Acknowledgements
  6. Author contributions statement
  7. Competing interests
  8. Appendix A: Derivation of the function $\mathcal{L}$
  9. Appendix B: Eigenvalue equation for the generic Bell operator

Key Result

Theorem 1

A linear congruence is a congruence relation of the form where $a,b,X,m \in \mathbb{Z}, a \neq 0, m > 0$. Let $g=\mathrm{gcd}(a,m)$. Then the linear congruence has a solution if and only if $b$ is divisible by $g$.

Figures (3)

  • Figure 1: Geometrical representation of the generic Bell inequalities for two measurements settings. (a) Four delta functions $\delta_d (a_1 + a_2 + a_3)$, $\delta_d (b_1 + a_2 + b_3+1)$, $\delta_d (b_1 + b_2 + a_3+1)$, and $\delta_d (a_1 + b_2 + b_3+1)$ represented by solid lines connecting variables $a_i$ and $b_j$ at each vertex. (b) Value assignment showing that at most three delta functions can be simultaneously satisfied. The variables in the blue lines indicate solutions satisfying the corresponding delta functions from the left diagram, while following the circular path clockwise leads to the red line which represents a constraint that cannot be satisfied, proving that all four delta function cannot be satisfied simultaneously. This geometric approach demonstrates the classical upper bound as 3$d$/4 - 1.
  • Figure 2: Geometrical representation for calculating the classical upper bound with three measurement settings. Hexagonal representation of the nine delta functions, with vertices representing variables and dotted lines connecting identical values. (a)-(c) Three distinct loops generate the constraints, confirming that at most seven delta functions can simultaneously equal unity.
  • Figure 3: Geometrical representation for calculating the classical upper bound with four measurement settings. All sixteen delta functions can be divided by four groups. As each group has the equivalent structure individually, the classical upper bound from one group is essential to calculate the total values of the classical upper bound.

Theorems & Definitions (1)

  • Theorem 1