Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Genuine monogamy relations in no-signaling theories - a geometric approach

Junghee Ryu, Daemin Lee, Jinhyoung Lee, Paweł Kurzyński, Dagomir Kaszlikowski

TL;DR

This paper develops a geometric, no-signaling framework for deriving monogamy relations of multipartite Bell inequalities. It uses statistical separation as a distance on probabilistic events and the triangle inequality to obtain Bell-type inequalities and various monogamy constraints, including genuine monogamies where only one inequality can be violated among many. The authors extend the approach to $d$-outcome scenarios via a quasi-distance, yielding Svetlichny-type tripartite inequalities and corresponding monogamies for qudits. The results tighten the constraints on nonlocal correlations beyond local realism and have potential applications in quantum cryptography and many-body physics, while also suggesting avenues for future work on higher-party and more general multipartite monogamies.

Abstract

Quantum correlations are subject to certain distribution rules represented by so-called monogamy relations. Derivation of monogamy relations for multipartite systems is a non-trivial problem, as the multipartite correlations reveal their behaviors in a way different from bipartite systems. We here show that simple geometric properties of probabilistic spaces, in conjunction with no-signaling principle, lead to genuine monogamy relations for a large class of Bell type inequalities for many qubits. The term of 'genuine' implies that only one out of $N$ Bell inequalities exhibits a quantum violation. We also generalize our method to qudits. Using the similar geometric approach with a quasi-distance employed, we derive Svetlichny-Zohren-Gill type Bell inequalities for $d$-dimensional tripartite systems, and show their monogamous nature.

Genuine monogamy relations in no-signaling theories - a geometric approach

TL;DR

This paper develops a geometric, no-signaling framework for deriving monogamy relations of multipartite Bell inequalities. It uses statistical separation as a distance on probabilistic events and the triangle inequality to obtain Bell-type inequalities and various monogamy constraints, including genuine monogamies where only one inequality can be violated among many. The authors extend the approach to -outcome scenarios via a quasi-distance, yielding Svetlichny-type tripartite inequalities and corresponding monogamies for qudits. The results tighten the constraints on nonlocal correlations beyond local realism and have potential applications in quantum cryptography and many-body physics, while also suggesting avenues for future work on higher-party and more general multipartite monogamies.

Abstract

Quantum correlations are subject to certain distribution rules represented by so-called monogamy relations. Derivation of monogamy relations for multipartite systems is a non-trivial problem, as the multipartite correlations reveal their behaviors in a way different from bipartite systems. We here show that simple geometric properties of probabilistic spaces, in conjunction with no-signaling principle, lead to genuine monogamy relations for a large class of Bell type inequalities for many qubits. The term of 'genuine' implies that only one out of Bell inequalities exhibits a quantum violation. We also generalize our method to qudits. Using the similar geometric approach with a quasi-distance employed, we derive Svetlichny-Zohren-Gill type Bell inequalities for -dimensional tripartite systems, and show their monogamous nature.

Paper Structure

This paper contains 18 sections, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Monogamy relations for many qubits
  3. Preliminaries
  4. Deriving Bell inequalities by using the triangle inequality
  5. Primary monogamy relation
  6. Genuine monogamies of four tripartite Bell inequalities
  7. Generalization to multipartite case
  8. $d$-outcome scenario
  9. Quasi-distance
  10. Bell inequalities by using the quasi-distance
  11. Monogamy relations
  12. Quantum violations
  13. Conclusions
  14. Kolmogorov probabilistic description
  15. Local model
  16. ...and 3 more sections

Figures (3)

  • Figure 1: (color online). No-signaling monogamy diagram for two tripartite Bell inequalities $\mathcal{B}_{ABC}^{111}$ (red) and $\mathcal{B}_{ABD}^{122}$ (blue). Only one of the inequalities can be violated by quantum theory.
  • Figure 2: (color online). Genuine monogamies for four tripartite Bell inequalities of $\mathcal{B}_{ABC}^{111}$ (red), $\mathcal{B}_{ABD}^{122}$ (blue), $\mathcal{B}_{ACD}^{212}$ (green), and $\mathcal{B}_{BCD}^{122}$ (black). (a) Genuine monogamy relation states that only one of the inequalities $\mathcal{B}_{ABC}^{111}, \mathcal{B}_{ABD}^{122}$, and $\mathcal{B}_{ACD}^{212}$ can be violated. (b) Full symmetric monogamy relation holds if we add one more Bell inequality $\mathcal{B}_{BCD}^{122}$.
  • Figure 3: Value of the left-hand side of the Bell inequality in (\ref{['EQ:3_ZG']}) as a function of the dimension $d$ up to $50$. As the lowerbound of the inequality by the local realistic description is zero, thus the negative values of the results imply the quantum violations.