A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving $(N+1)$ -partite CHSH-type Bell inequalities from $N$ -partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving $(N+1)$-partite CHSH-type Bell inequalities from $N$-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.

arXiv:1302.6698Physical Review AAug 29, 201311 citations

On the CHSH Form of Bell’s Inequalities

Justo Pastor Lambare

Foundations of PhysicsJan 19, 20179 citations

Clauser-Horne-Shimony-Holt-type Bell inequalities involving a party with two or three local binary settings

Yu-Chun Wu, Piotr Badziąg, Marek Żukowski

We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.

arXiv:0810.1169Physical Review AFeb 11, 20098 citations

A Quantum Mechanical Bound for CHSH-Type Bell Inequalities

Michael Epping, Hermann Kampermann, Dagmar Bruß

arXiv:1501.01406The frontiers collectionNov 15, 20161 citation

Quantum sets of the multicolored-graph approach to contextuality

Lina Vandré, Marcelo Terra Cunha

The Clauser-Horne-Shimony-Holt (CHSH) inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the Cabello-Severini-Winter (CSW) approach, one may think that the theta body of the CHSH graph $\mathrm{TH}({G}_{\text{CHSH}})$ is equal to the quantum set of the CHSH Bell inequality ${\mathcal{Q}}_{\text{CHSH}}$ , but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while the CSW approach only demands the exclusivity structure of a noncontextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like the CHSH one, the colored-graph approach was introduced. Does it make any difference to think about the CHSH scenario as a Bell one or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph $\mathcal{G}$ and the NC CHSH inequality by a simple graph $G$ , which is the shadow of the colored graph $\mathcal{G}$ . In general, we have that the theta body of the colored graph ${\ensuremath{\theta}}_{c}(\mathcal{G})$ is a subset of the theta body of its shadow graph $\mathrm{TH}(G)$ in the same way that the Lov\'asz number, which corresponds to the quantum bound, of the simple graph $\ensuremath{\vartheta}(G)$ is greater than or equal to the Lov\'asz number of the colored graph ${\ensuremath{\vartheta}}_{c}(\mathcal{G})$. In the case of the CHSH inequality, we have that $\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})$. Does this accident also hold for the corresponding quantum sets? Is it true that $\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})$, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negative. We show that ${\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)$ and therefore that there are quantum correlations which cannot be obtained under Bell restrictions.The Clauser-Horne-Shimony-Holt (CHSH) inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the Cabello-Severini-Winter (CSW) approach, one may think that the theta body of the CHSH graph $\mathrm{TH}({G}_{\text{CHSH}})$ is equal to the quantum set of the CHSH Bell inequality ${\mathcal{Q}}_{\text{CHSH}}$, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while the CSW approach only demands the exclusivity structure of a noncontextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like the CHSH one, the colored-graph approach was introduced. Does it make any difference to think about the CHSH scenario as a Bell one or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph $\mathcal{G}$ and the NC CHSH inequality by a simple graph $G$, which is the shadow of the colored graph $\mathcal{G}$. In general, we have that the theta body of the colored graph ${\ensuremath{\theta}}_{c}(\mathcal{G})$ is a subset of the theta body of its shadow graph $\mathrm{TH}(G)$ in the same way that the Lov\'asz number, which corresponds to the quantum bound, of the simple graph $\ensuremath{\vartheta}(G)$ is greater than or equal to the Lov\'asz number of the colored graph ${\ensuremath{\vartheta}}_{c}(\mathcal{G})$. In the case of the CHSH inequality, we have that $\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})$. Does this accident also hold for the corresponding quantum sets? Is it true that $\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})$, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negative. We show that ${\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)$ and therefore that there are quantum correlations which cannot be obtained under Bell restrictions.

arXiv:2105.08561Physical Review ADec 16, 20225 citations

SYSTEMATIC CONSTRUCTION OF GENERALIZED BELL INEQUALITIES

SHOGO TANIMURA

Jul 16, 20120 citations

(N+1)

-partite CHSH-type Bell inequalities from

N

-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.

\mathrm{TH}({G}_{\text{CHSH}})

is equal to the quantum set of the CHSH Bell inequality

{\mathcal{Q}}_{\text{CHSH}}

, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while the CSW approach only demands the exclusivity structure of a noncontextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like the CHSH one, the colored-graph approach was introduced. Does it make any difference to think about the CHSH scenario as a Bell one or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph

\mathcal{G}

and the NC CHSH inequality by a simple graph

G

, which is the shadow of the colored graph

\mathcal{G}

. In general, we have that the theta body of the colored graph ${\ensuremath{\theta}}_{c}(\mathcal{G})$ is a subset of the theta body of its shadow graph

\mathrm{TH}(G)

in the same way that the Lov\'asz number, which corresponds to the quantum bound, of the simple graph $\ensuremath{\vartheta}(G)$ is greater than or equal to the Lov\'asz number of the colored graph ${\ensuremath{\vartheta}}_{c}(\mathcal{G})$. In the case of the CHSH inequality, we have that $\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})$. Does this accident also hold for the corresponding quantum sets? Is it true that $\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})$, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negative. We show that ${\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)$ and therefore that there are quantum correlations which cannot be obtained under Bell restrictions.