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

arXiv:quant-ph/0512100

Extremal quantum correlations for N parties with two dichotomic observables per site

Lluís Masanes

Published Dec 13, 2005• 45 citations
arXivHTMLPDF

Abstract

Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring $N$-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an $N$-partite state $ρ$ violates a given Bell inequality, then, $ρ$ can be transformed by stochastic local operations into an $N$-qubit state that violates the same Bell inequality by an equal or larger amount.

DOI
10.48550/arxiv.quant-ph/0512100

Related papers

All-multipartite Bell-correlation inequalities for two dichotomic observables per site

Reinhard F. Werner, Michael M. Wolf

We construct a set of ${2}^{{2}^{n}}$ independent Bell-correlation inequalities for n-partite systems with two dichotomic observables each, which is complete in the sense that the inequalities are satisfied if and only if the correlations considered allow a local classical model. All these inequalities can be summarized in a single, albeit nonlinear inequality. We show that quantum correlations satisfy this condition provided the state has positive partial transpose with respect to any grouping of the n systems into two subsystems. We also provide an efficient algorithm for finding the maximal quantum-mechanical violation of each inequality, and show that the maximum is always attained for the generalized GHZ state.

arXiv:quant-ph/0102024•Physical Review A•Aug 17, 2001•497 citations

Dualistic operational characterization of device-dependent correlation sets via convex analysis in the $(2,m,2)$ Bell scenario

Ryosuke Nogami, Jaeha Lee

We analyze device-dependent correlation sets generated by fixed local dichotomic measurements for two-qubit systems in the $(2,m,2)$ Bell scenario. We consider three fundamental state spaces for the composite system: the separable state space, the standard quantum state space, and the maximal tensor-product state space, which contains beyond-quantum states compatible with local quantum measurements. We formulate the corresponding correlation sets for general fixed dichotomic measurements and, in the traceless case, derive particularly simple explicit formulae for their support and gauge functions. These functions furnish dual operational characterizations of the three correlation sets: the support functions give optimal witnesses for entanglement and beyond-quantum states, whereas the gauge functions quantify the robustness of these detections against depolarizing noise. We further derive convex-hull representations that elucidate the extremal structures of the correlation sets and the physical states realizing them, showing in particular that extremal quantum correlations are realized by maximally entangled states. The fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions available to Alice and Bob. When both parties have three linearly independent measurement directions, our entanglement criterion detects Werner states up to the optimal PPT threshold $p_{\mathrm{crit}}=2/3$. For beyond-quantum-state detection, a nontrivial separation from the quantum set occurs only under the same measurement condition; in that case, the same optimal noise threshold is attained for an extremal state in the maximal tensor-product state space.

quant-pharXiv:2604.24423•Apr 28, 2026•0 citations

Small violations of full-correlation Bell inequalities for multipartite pure random states

Raphael C. Drumond, Roberto I. Oliveira

We estimate the probability of random $N$-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with $N$, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with $N$ is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing any violation is already small. For the qubits case we discuss, furthermore, the consequences of this result for the probability of seeing arbitrary violations (i.e., of any order of magnitude) when experimental imperfections are considered.

arXiv:1209.1755•Physical Review A•Jul 24, 2012•7 citations

Geometric structure of quantum correlators via semidefinite programming

Lê Phuc Thinh, Antonios Varvitsiotis, Yu Cai

Quantum information leverages correlations between spacelike separated parties in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum correlators. In this paper we study the structure of the set of quantum correlators using semidefinite programming, more precisely the boundary, extreme, and exposed points. We obtain quantum Bell inequalities characterizing a certain class of bipartite scenarios. In the case of two dichotomic measurements, extremal quantum correlators coincide with the correlators that uniquely determine the state and measurement operators, a property known as self-testing. We illustrate the usefulness of our theoretical findings with many examples and extensive computational work.

arXiv:1809.10886•Physical Review A•May 10, 2019•11 citations

Quantum correlations on the no-signaling boundary: self-testing and more

Kai-Siang Chen, Gelo Noel M. Tabia, Chellasamy Jebarathinam +3 more

In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k) -parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.

arXiv:2207.13850•Quantum•Jul 11, 2023•12 citations

Maximal qubit violations of n -locality in star and chain networks

Brian Doolittle, Eric Chitambar

The nonlocal correlations of noisy quantum systems are important for both understanding nature and developing quantum technology. We consider the correlations of star and chain quantum networks in which noisy entanglement sources are measured by nonsignaling parties. We derive the necessary and sufficient conditions for when a broad family of star and chain $n$-locality inequalities achieve their maximal quantum violation assuming that each party's dichotomic observables are separable across qudit systems. When pairs of local dichotomic observables are considered on qubit systems, we derive maximal $n$-local violations that are larger and more robust to noise than the maximal $n$-locality violations reported previously. To obtain these larger values, we consider observables that we have not found in previous studies. Thus, we gain insights into self-testing entanglement sources and measurements in star and chain networks.

arXiv:2212.06915•Physical Review A•Oct 9, 2023•11 citations
Consider a scenario where NNN separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the NNN events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring NNN-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an NNN-partite state ρρρ violates a given Bell inequality, then, ρρρ can be transformed by stochastic local operations into an NNN-qubit state that violates the same Bell inequality by an equal or larger amount.
We construct a set of 22n{2}^{{2}^{n}}22n independent Bell-correlation inequalities for n-partite systems with two dichotomic observables each, which is complete in the sense that the inequalities are satisfied if and only if the correlations considered allow a local classical model. All these inequalities can be summarized in a single, albeit nonlinear inequality. We show that quantum correlations satisfy this condition provided the state has positive partial transpose with respect to any grouping of the n systems into two subsystems. We also provide an efficient algorithm for finding the maximal quantum-mechanical violation of each inequality, and show that the maximum is always attained for the generalized GHZ state.
Dualistic operational characterization of device-dependent correlation sets via convex analysis in the (2,m,2)(2,m,2)(2,m,2) Bell scenario
We analyze device-dependent correlation sets generated by fixed local dichotomic measurements for two-qubit systems in the (2,m,2)(2,m,2)(2,m,2) Bell scenario. We consider three fundamental state spaces for the composite system: the separable state space, the standard quantum state space, and the maximal tensor-product state space, which contains beyond-quantum states compatible with local quantum measurements. We formulate the corresponding correlation sets for general fixed dichotomic measurements and, in the traceless case, derive particularly simple explicit formulae for their support and gauge functions. These functions furnish dual operational characterizations of the three correlation sets: the support functions give optimal witnesses for entanglement and beyond-quantum states, whereas the gauge functions quantify the robustness of these detections against depolarizing noise. We further derive convex-hull representations that elucidate the extremal structures of the correlation sets and the physical states realizing them, showing in particular that extremal quantum correlations are realized by maximally entangled states. The fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions available to Alice and Bob. When both parties have three linearly independent measurement directions, our entanglement criterion detects Werner states up to the optimal PPT threshold pcrit=2/3p_{\mathrm{crit}}=2/3pcrit​=2/3. For beyond-quantum-state detection, a nontrivial separation from the quantum set occurs only under the same measurement condition; in that case, the same optimal noise threshold is attained for an extremal state in the maximal tensor-product state space.
We estimate the probability of random NNN-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with NNN, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with NNN is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing any violation is already small. For the qubits case we discuss, furthermore, the consequences of this result for the probability of seeing arbitrary violations (i.e., of any order of magnitude) when experimental imperfections are considered.
Quantum information leverages correlations between spacelike separated parties in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum correlators. In this paper we study the structure of the set of quantum correlators using semidefinite programming, more precisely the boundary, extreme, and exposed points. We obtain quantum Bell inequalities characterizing a certain class of bipartite scenarios. In the case of two dichotomic measurements, extremal quantum correlators coincide with the correlators that uniquely determine the state and measurement operators, a property known as self-testing. We illustrate the usefulness of our theoretical findings with many examples and extensive computational work.
In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k) -parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.
The nonlocal correlations of noisy quantum systems are important for both understanding nature and developing quantum technology. We consider the correlations of star and chain quantum networks in which noisy entanglement sources are measured by nonsignaling parties. We derive the necessary and sufficient conditions for when a broad family of star and chain nnn-locality inequalities achieve their maximal quantum violation assuming that each party's dichotomic observables are separable across qudit systems. When pairs of local dichotomic observables are considered on qubit systems, we derive maximal nnn-local violations that are larger and more robust to noise than the maximal nnn-locality violations reported previously. To obtain these larger values, we consider observables that we have not found in previous studies. Thus, we gain insights into self-testing entanglement sources and measurements in star and chain networks.