Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

General Iterative Formulas of Bell's Inequalities.

Xing-Yan Fan, Hongye Liu, Jia-Le Miao, Yi-Jia Liu, Wei-Min Shang, Jie Zhou, Hui-Xian Meng, Jing‐Ling Chen

Published Sep 12, 2021• 0 citations
OpenAlex

Abstract

Newton's second law aids us in predicting the location of a classical object after knowing its initial position and velocity together with the force it experiences at any time, which can be seen as a process of continuous iteration. When it comes to discrete problems, e.g. building Bell inequalities, as a vital tool to study the powerful nonlocal correlations in quantum information processing. Unless having known precisely the general formula of associated inequalities, iterative formulas build a bridge from simple examples to all elements in the set of Bell inequalities. Although exhaust all entities in the set, even in the subset of tight individuals, is a NP hard problem, it is possible to find out the evolution law of Bell inequalities from few-body, limited-setting and low-dimension situations to arbitrary $(n,k,d)$ constructions, i.e. $n$ particles, $k$ measurements per particle, and $d$ outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the root method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all $(n,k,2)$ circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining root method on the one hand, and explore more interesting inequalities on the other.

Related papers

Generalizing Optimal Bell Inequalities

Fabian Bernards, Otfried Gühne

Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows one to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particle inequality by Froissart [Nuovo Cimento Soc. Ital. Fis. B 64, 241 (1981)], also known as I3322 inequality, to three particles. For the simplest of these inequalities, we study their quantum mechanical properties and demonstrate that they are relevant, in the sense that they detect nonlocality of quantum states, for which all two-setting inequalities fail to do so.

arXiv:2005.08687•Physical Review Letters•Nov 13, 2020•7 citations

Generalized iterative formula for Bell inequalities

Xing-Yan Fan, Zhen‐Peng Xu, Jia-Le Miao +7 more

Bell inequality is a vital tool to detect the nonlocal correlations, but the construction of it for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of $(n+1)$-partite Bell inequalities into $n$-partite ones, we present a generalized iterative formula to construct nontrivial $(n+1)$-partite ones from the $n$-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinsk\ifmmode \breve{\imath}\else \u{\i}\fi{}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the nonlocality in the entire entangled region. Furthermore, we present a generalization of the I3322 inequality to any $n$-partite case which is still tight, and of the 46 \ifmmode \acute{S}\else \'{S}\fi{}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.

arXiv:2109.05521•Physical Review A•Dec 6, 2023•1 citation

Studies On Bell's Theorem

V. Uğur Güney

In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible quantities are not allowed by classical mechanics, a feature that is called as "quantum nonlocality". An experimental observation of those correlations, in other words, a violation of the limits imposed by classical physics, implies the correctness of quantum description and invalidates the classical, local realistic models. The first Bell experiments were proposed by Clauser, Horne, Shimony, and Holt, who invented the most famous Bell's inequality [13]. Later, the Aspect experiments were satisfactory enough for the physics community to be conclusive about the validation of quantum mechanics [1][3][4][2]. Ekert's work on applications of quantum nonlocality to communication resulted in the new field of quantum communication and cryptography, and turned the research program into a practical one [20]. Pitowsky showed a method to find all expressions of limitations due to local realism, all Bell's inequalities, for a given physical scenario. He also proved that the problem is, unfortunately, NP-complete and hence as the scenarios get more complex, they also become computationally intractable [33][34]. Therefore, different methods for the solution of special cases of the problem are necessary. Inequalities found for those special cases can be called classes of Bell's inequalities. For example, Werner and Wolf [41] and Collins, Gisin, Linden, Massar, and Popescu [16] found classes that cover a wide range of scenarios. Our work is a similar kind of effort to produce and study new types of Bell's inequalities.

•Jan 1, 2015•0 citations

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.

quant-pharXiv:2602.08469•Feb 10, 2026•0 citations

Avenues to generalising Bell inequalities

Marcin Karczewski, Giovanni Scala, Antonio Mandarino +2 more

Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.

arXiv:2202.06606•Journal of Physics A Mathematical and Theoretical•Aug 16, 2022•10 citations

Bell inequalities and large-scale quantum networks

Michael Epping

In the field of quantum information theory one investigates the properties of information carriers which are subject to the laws of quantum mechanics. The differences between quantum mechanics and the laws that govern macroscopic information carriers lead to new opportunities and challenges in information processing. One of these differences is the entanglement of particles, which leads to strong correlations of measurement outcomes. This allows the violation of so-called Bell inequalities in quantum mechanics, which is impossible in ``classical'' theories. I investigated the possible amount of violation for an important class of Bell inequalities. In doing so I found a simple mathematical expression for an upper bound and studied the achievability of that bound. The approach provides a basic understanding of the considered Bell inequalities, which allows to construct new inequalities with interesting properties. In particular one can understand how Bell inequalities allow to bound the dimension of a quantum system. Changing a Bell inequality with invariant quantum value proved useful in optimizing them with respect to the violation. The distribution of entangled systems across large distances is achieved by sending photons. However, since they are absorbed in long fibers, quantum repeater become necessary for distances larger than approximately 200 km. Several approaches to counter the losses are known. In the long term error correction codes are very promising. Here the information is encoded into many photons, such that some losses can be compensated. I contributed to the analysis of this approach by showing how it naturally generalizes to networks of quantum repeaters. The formalism of graph states is useful in this context. A complete performance analysis of a quantum repeater contains many sources of errors and their propagation inside the circuit. I extended this analysis compared to the literature. At the moment it is not clear which types of quantum repeaters will prevail. My comparison of different theoretical proposals helps towards answering this question.

•Jan 1, 2016•0 citations
Newton's second law aids us in predicting the location of a classical object after knowing its initial position and velocity together with the force it experiences at any time, which can be seen as a process of continuous iteration. When it comes to discrete problems, e.g. building Bell inequalities, as a vital tool to study the powerful nonlocal correlations in quantum information processing. Unless having known precisely the general formula of associated inequalities, iterative formulas build a bridge from simple examples to all elements in the set of Bell inequalities. Although exhaust all entities in the set, even in the subset of tight individuals, is a NP hard problem, it is possible to find out the evolution law of Bell inequalities from few-body, limited-setting and low-dimension situations to arbitrary (n,k,d)(n,k,d)(n,k,d) constructions, i.e. nnn particles, kkk measurements per particle, and ddd outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the root method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all (n,k,2)(n,k,2)(n,k,2) circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining root method on the one hand, and explore more interesting inequalities on the other.
Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows one to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particle inequality by Froissart [Nuovo Cimento Soc. Ital. Fis. B 64, 241 (1981)], also known as I3322 inequality, to three particles. For the simplest of these inequalities, we study their quantum mechanical properties and demonstrate that they are relevant, in the sense that they detect nonlocality of quantum states, for which all two-setting inequalities fail to do so.
Bell inequality is a vital tool to detect the nonlocal correlations, but the construction of it for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of (n+1)(n+1)(n+1)-partite Bell inequalities into nnn-partite ones, we present a generalized iterative formula to construct nontrivial (n+1)(n+1)(n+1)-partite ones from the nnn-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinsk\ifmmode \breve{\imath}\else \u{\i}\fi{}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the nonlocality in the entire entangled region. Furthermore, we present a generalization of the I3322 inequality to any nnn-partite case which is still tight, and of the 46 \ifmmode \acute{S}\else \'{S}\fi{}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.
In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible quantities are not allowed by classical mechanics, a feature that is called as "quantum nonlocality". An experimental observation of those correlations, in other words, a violation of the limits imposed by classical physics, implies the correctness of quantum description and invalidates the classical, local realistic models. The first Bell experiments were proposed by Clauser, Horne, Shimony, and Holt, who invented the most famous Bell's inequality [13]. Later, the Aspect experiments were satisfactory enough for the physics community to be conclusive about the validation of quantum mechanics [1][3][4][2]. Ekert's work on applications of quantum nonlocality to communication resulted in the new field of quantum communication and cryptography, and turned the research program into a practical one [20]. Pitowsky showed a method to find all expressions of limitations due to local realism, all Bell's inequalities, for a given physical scenario. He also proved that the problem is, unfortunately, NP-complete and hence as the scenarios get more complex, they also become computationally intractable [33][34]. Therefore, different methods for the solution of special cases of the problem are necessary. Inequalities found for those special cases can be called classes of Bell's inequalities. For example, Werner and Wolf [41] and Collins, Gisin, Linden, Massar, and Popescu [16] found classes that cover a wide range of scenarios. Our work is a similar kind of effort to produce and study new types of Bell's inequalities.
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)(2,2,2)(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)(2,2,3)(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.
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.
In the field of quantum information theory one investigates the properties of information carriers which are subject to the laws of quantum mechanics. The differences between quantum mechanics and the laws that govern macroscopic information carriers lead to new opportunities and challenges in information processing. One of these differences is the entanglement of particles, which leads to strong correlations of measurement outcomes. This allows the violation of so-called Bell inequalities in quantum mechanics, which is impossible in ``classical'' theories. I investigated the possible amount of violation for an important class of Bell inequalities. In doing so I found a simple mathematical expression for an upper bound and studied the achievability of that bound. The approach provides a basic understanding of the considered Bell inequalities, which allows to construct new inequalities with interesting properties. In particular one can understand how Bell inequalities allow to bound the dimension of a quantum system. Changing a Bell inequality with invariant quantum value proved useful in optimizing them with respect to the violation. The distribution of entangled systems across large distances is achieved by sending photons. However, since they are absorbed in long fibers, quantum repeater become necessary for distances larger than approximately 200 km. Several approaches to counter the losses are known. In the long term error correction codes are very promising. Here the information is encoded into many photons, such that some losses can be compensated. I contributed to the analysis of this approach by showing how it naturally generalizes to networks of quantum repeaters. The formalism of graph states is useful in this context. A complete performance analysis of a quantum repeater contains many sources of errors and their propagation inside the circuit. I extended this analysis compared to the literature. At the moment it is not clear which types of quantum repeaters will prevail. My comparison of different theoretical proposals helps towards answering this question.