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Bell Inequalities for Smells

Ricardo Faleiro, Flavien Hirsch, Emmanuel Zambrini Cruzeiro, Nicolas Gisin

Abstract

In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. This arises naturally when outcomes are difficult to characterize. For instance, if measurements yield smells, it may be impractical to process them individually, while still being reasonable to judge whether two smells are identical or not. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator inequalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural properties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.

Bell Inequalities for Smells

Abstract

In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. This arises naturally when outcomes are difficult to characterize. For instance, if measurements yield smells, it may be impractical to process them individually, while still being reasonable to judge whether two smells are identical or not. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator inequalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural properties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.
Paper Structure (15 sections, 10 theorems, 83 equations, 2 figures, 6 tables)

This paper contains 15 sections, 10 theorems, 83 equations, 2 figures, 6 tables.

Table of Contents

  1. Local strategies in Bell inequalities for smells and their saturating bound
  2. Proof of Theorem \ref{['Th1']}
  3. Bipartite case
  4. Multipartite case
  5. Construction of unanimous vertices
  6. Bell inequalities for smells: A partial repository
  7. Bell facets for smells/unanimous in $(2,m,k)$
  8. Multipartite Bell facets for smells
  9. Multipartite Unanimous Bell Inequalities
  10. Genuine multipartite nonlocality
  11. All unanimous inequalities are nonlocal games
  12. Unanimous multipartite family: Proof of local bound and some quantum violations
  13. No-signaling vertices of the unanimous scenario
  14. Bipartite scenarios
  15. Multipartite scenario

Key Result

Theorem 1

Consider an $(n,m,k)$ Bell scenario. There exists a finite integer $k^*$ such that the local polytope $\mathcal{L}_\Sigma(n,m,k)$ coincides with $\mathcal{L}_\Sigma(n,m,k^*)$ for all $k \geq k^*$. The explicit expression is given by that is, where $\lfloor . \rfloor$ is the floor function.

Figures (2)

  • Figure 1: A strategy maximizing the number of outputs for $3$ parties with $3$ inputs each, using cyclic procedure \ref{['cyclic_procedure']}. We end up with $3$ isolated nodes and a perfect matching of $3$ pairs for the remaining nodes, reaching $k^* = 6$.
  • Figure 2: A strategy maximizing the number of outputs for $3$ parties with $4$ inputs each, using cyclic procedure \ref{['cyclic_procedure']}. We end up with $4$ isolated nodes and a perfect matching of $4$ pairs for the remaining nodes, reaching $k^* = 7$. Note that the node with dashes around is "unused" due to the fact that both $n$ and $m-1$ are odd.

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 5 more