Violation of Bell Inequalities for Multi-Parties

outcomes, the Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements, but the quantum bound is dependent both on

n

and

d

. Even so, when

d

is large enough, the quantum bound is only dependent on

d

. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. All the corresponding nonlocal games are analyzed.

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Violation of Bell inequalities for multipartite systems

Yanmin Yang, Zhu‐Jun Zheng

In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$ , but the quantum bound is dependent both on $n$ and $d$ . Even so, when $d$ is large enough, the quantum bound is only dependent on $d$ . The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed.In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$, but the quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large enough, the quantum bound is only dependent on $d$. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed.

arXiv:1705.00513May 1, 20170 citations

Bell inequalities from group actions of single-generator groups

V. Uğur Güney, Mark S. Hillery

We study a method of generating Bell inequalities by using group actions of single-generator Abelian groups. Two parties, Alice and Bob, each make one of $M$ possible measurements on a system, with each measurement having $K$ possible outcomes. The probabilities for the outcomes of these measurements are $P({a}_{j}=k,{b}_{{j}^{\ensuremath{'}}}={k}^{\ensuremath{'}})$, where $j,{j}^{\ensuremath{'}}\ensuremath{\in}{1,2,...,M}$ and $k,{k}^{\ensuremath{'}}\ensuremath{\in}{0,1,...,K\ensuremath{-}1}$. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including $M=2$ , $K=3$ , and general $M$ , $K=2$ . We also discuss the resulting inequalities in terms of nonlocal games.We study a method of generating Bell inequalities by using group actions of single-generator Abelian groups. Two parties, Alice and Bob, each make one of $M$ possible measurements on a system, with each measurement having $K$ possible outcomes. The probabilities for the outcomes of these measurements are $P({a}_{j}=k,{b}_{{j}^{\ensuremath{'}}}={k}^{\ensuremath{'}})$, where $j,{j}^{\ensuremath{'}}\ensuremath{\in}{1,2,...,M}$ and $k,{k}^{\ensuremath{'}}\ensuremath{\in}{0,1,...,K\ensuremath{-}1}$. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including $M=2$, $K=3$, and general $M$, $K=2$. We also discuss the resulting inequalities in terms of nonlocal games.

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V. Uğur Güney, Mark S. Hillery

In a previous publication, we showed how group actions can be used to generate Bell inequalities. The group action yields a set of measurement probabilities whose sum is the basic element in the inequality. The sum has an upper bound if the probabilities are a result of a local, realistic theory, but this bound can be violated if the probabilities come from quantum mechanics. In our previous paper, we considered the case of only two parties making the measurements and single-generator groups. Here we show that the method can be extended to three parties, and it can also be extended to non-Abelian groups. We discuss the resulting inequalities in terms of nonlocal games.In a previous publication, we showed how group actions can be used to generate Bell inequalities. The group action yields a set of measurement probabilities whose sum is the basic element in the inequality. The sum has an upper bound if the probabilities are a result of a local, realistic theory, but this bound can be violated if the probabilities come from quantum mechanics. In our previous paper, we considered the case of only two parties making the measurements and single-generator groups. Here we show that the method can be extended to three parties, and it can also be extended to non-Abelian groups. We discuss the resulting inequalities in terms of nonlocal games.

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Violation of Bell inequalities from $S_4$ S 4 symmetry: the three orbits caseViolation of Bell inequalities from $$S_4$$ S 4 symmetry: the three orbits case

Katarzyna Bolonek-Lasoń, Ścibór Sobieski

The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to $S_4$ group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of $S_4$ are derived and their violation is described. The corresponding nonlocal games are analyzed.The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to $$S_4$$ group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of $$S_4$$ are derived and their violation is described. The corresponding nonlocal games are analyzed.

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Violation of Bell inequality based on S4 symmetry

Katarzyna Bolonek-Lasoń

In two recent papers [Phys. Rev. A 90, 062121 (2014) and Phys. Rev. A, 91, 052110 (2015)] an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. Here we apply this method to more complicated examples of ${S}_{4}$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of ${S}_{4}$ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.In two recent papers [Phys. Rev. A 90, 062121 (2014) and Phys. Rev. A, 91, 052110 (2015)] an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. Here we apply this method to more complicated examples of ${S}_{4}$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of ${S}_{4}$ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.

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Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

Palash Pandya, Shubhayan Sarkar, Remigiusz Augusiak

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.

quant-pharXiv:2602.08469Feb 10, 20260 citations

d

possible measurements, each measurement has

n

outcomes, the Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements, but the quantum bound is dependent both on

n

and

d

. Even so, when

d

is large enough, the quantum bound is only dependent on

d

d

possible measurements, each measurement has

n

outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements

d

, but the quantum bound is dependent both on

n

and

d

. Even so, when

d

is large enough, the quantum bound is only dependent on

d

We study a method of generating Bell inequalities by using group actions of single-generator Abelian groups. Two parties, Alice and Bob, each make one of

M

possible measurements on a system, with each measurement having

K

possible outcomes. The probabilities for the outcomes of these measurements are $P({a}_{j}=k,{b}_{{j}^{\ensuremath{'}}}={k}^{\ensuremath{'}})$, where $j,{j}^{\ensuremath{'}}\ensuremath{\in}{1,2,...,M}$ and $k,{k}^{\ensuremath{'}}\ensuremath{\in}{0,1,...,K\ensuremath{-}1}$. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including

M=2

K=3

, and general

M

K=2

. We also discuss the resulting inequalities in terms of nonlocal games.

Violation of Bell inequalities from

S_4

S 4 symmetry: the three orbits case

The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to

S_4

group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of

S_4

are derived and their violation is described. The corresponding nonlocal games are analyzed.

{S}_{4}

symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of

{S}_{4}

determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.

(2,2,2)

scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the

(2,2,3)

scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.