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:1111.0822 · Physical Review A

Optimal measurement bases for Bell tests based on the Clauser-Horne inequality

G. Lima, Enrique B. Inostroza, Reinaldo O. Vianna, Jan-Åke Larsson, C. Saavedra

Published Jan 19, 2012• 11 citations
arXivHTMLPDF

Abstract

The Hardy test of nonlocality can be seen as a particular case of the Bell tests based on the Clauser-Horne (CH) inequality. Here we stress this connection when we analyze the relation between the CH-inequality violation, its threshold detection efficiency, and the measurement settings adopted in the test. It is well known that the threshold efficiencies decrease when one considers partially entangled states and that the use of these states, unfortunately, generates a reduction in the CH violation. Nevertheless, these quantities are both dependent on the measurement settings considered, and in this paper we show that there are measurement bases which allow for an optimal situation in this trade-off relation. These bases are given as a generalization of the Hardy measurement bases, and they will be relevant for future Bell tests relying on pairs of entangled qubits.

DOI
10.1103/physreva.85.012105

Related papers

More nonlocality with less entanglement in Clauser-Horne-Shimony-Holt experiments using inefficient detectors

Daniel Dilley, Eric Chitambar

It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches $50%$ from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.

arXiv:1803.07747•Physical Review A•Jun 8, 2018•8 citations

Class of Bell-Clauser-Horne inequalities for testing quantum nonlocality

Chen Qian, Yang-Guang Yang, Qun Wang +1 more

Quantum nonlocality, one of the most important features of quantum mechanics, is normally connected in experiments with the violation of Bell-Clauser-Horne (Bell-CH) inequalities. We propose effective methods for the rearrangement and linear inequality to prove a large variety of Bell-CH inequalities. We also derive a set of Bell-CH inequalities using these methods which can be violated in some quantum entangled states.

arXiv:2101.02426•Physical Review A•Jun 1, 2021•4 citations

Bounding the detection efficiency threshold in Bell tests using multiple copies of the two-qubit maximally entangled state

István Márton, Erika Bene, Tamás Vértesi

In this paper we investigate the critical efficiency of detectors in order to see Bell nonlocality using multiple copies of the two-qubit maximally entangled state and local Pauli measurements which act in the corresponding qubit subspaces. It is known that for the two-qubit maximally entangled state a symmetric detection efficiency of $82.84\%$ can be tolerated using the Clauser-Horne-Shimony-Holt (CHSH) Bell test. We show that this threshold can be lowered by using multiple copies of the two-qubit maximally entangled state. We get the upper bounds $80.86\%$, $73.99\%$ and $69.29\%$ on the symmetric detection efficiency threshold for two, three and four copies of the state, where the respective number of measurements per party are 4, 8 and 16. However, in the case of four copies the result is partly due to a heuristic method. In order to get the corresponding Bell inequalities we made use of linear programming for two copies of the state and convex optimization based on Gilbert algorithm for three and four copies of the state.

•Mar 18, 2021•1 citation

Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles

István Márton, Erika Bene, Tamás Vértesi

In this paper, we investigate the critical efficiency of detectors to observe Bell nonlocality using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles, such as hyperentangled states, and the product of Pauli measurements. It is known that in a Clauser-Horne-Shimony-Holt (CHSH) Bell test the symmetric detection efficiency of $82.84%$ can be tolerated for the two-qubit maximally entangled state. We beat this enigmatic threshold by entangling two particles with multiple degrees of freedom. The obtained upper bounds of the symmetric detection efficiency thresholds are $80.86%$, $73.99%$, and $69.29%$ for two, three, and four copies of the two-qubit maximally entangled state, respectively. The number of measurements and outcomes in the respective cases are 4, 8, and 16. To find the improved thresholds, we use large-scale convex optimization tools, which allows us to significantly go beyond state-of-the-art results. The proof is exact up to three copies, while for four copies it is due to reliable numerical computations. Specifically, we used linear programming to obtain the two-copy threshold and the corresponding Bell inequality, and convex optimization based on Gilbert's algorithm for three and four copies of the two-qubit state. We show analytically that the symmetric detection efficiency threshold decays exponentially with the number of copies of the two-qubit state. Our techniques can also be applied to more general Bell nonlocality scenarios with more than two parties.

arXiv:2103.10413•Physical Review A•Feb 6, 2023•6 citations

Higher quantum bound for the Vértesi-Bene-Bell inequality and the role of positive operator-valued measures regarding its threshold detection efficiency

Johanna F. Barra, Esteban S. Gómez, Gustavo Cañas +3 more

Recently, V\'ertesi and Bene [Phys. Rev. A 82, 062115 (2010)] derived a two-qubit Bell inequality ${I}_{\mathrm{CH}3}$, which they show to be maximally violated only when more general positive operator-valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element POVM involved in the Bell test and obtain a higher quantum bound for the ${I}_{\mathrm{CH}3}$ inequality. With a higher quantum bound for ${I}_{\mathrm{CH}3}$, we investigate whether there is an experimental setup that can be used for observing that POVMs give higher violations in Bell tests based on this inequality. We analyze the maximum errors supported by the inequality to identify a source of entangled photons that can be used for the test. Then we study whether POVMs are also relevant in the more realistic case where partially entangled states are used in the experiment. Finally, we investigate the required efficiencies of the ${I}_{\mathrm{CH}3}$ inequality, and the type of measurements involved, for closing the detection loophole. We obtain the result that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal required detection efficiency (in the case of two qubits) of the Clauser-Horne-Bell inequality.

arXiv:1207.0712•Physical Review A•Oct 15, 2012•4 citations

Interplay among entanglement, measurement incompatibility, and nonlocality

Yuwei Zhu, Xingjian Zhang, Xiongfeng Ma

Abstract Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.

arXiv:2303.08407•Quantum Science and Technology•Jun 21, 2024•7 citations
The Hardy test of nonlocality can be seen as a particular case of the Bell tests based on the Clauser-Horne (CH) inequality. Here we stress this connection when we analyze the relation between the CH-inequality violation, its threshold detection efficiency, and the measurement settings adopted in the test. It is well known that the threshold efficiencies decrease when one considers partially entangled states and that the use of these states, unfortunately, generates a reduction in the CH violation. Nevertheless, these quantities are both dependent on the measurement settings considered, and in this paper we show that there are measurement bases which allow for an optimal situation in this trade-off relation. These bases are given as a generalization of the Hardy measurement bases, and they will be relevant for future Bell tests relying on pairs of entangled qubits.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
In this paper we investigate the critical efficiency of detectors in order to see Bell nonlocality using multiple copies of the two-qubit maximally entangled state and local Pauli measurements which act in the corresponding qubit subspaces. It is known that for the two-qubit maximally entangled state a symmetric detection efficiency of 82.84%82.84\%82.84% can be tolerated using the Clauser-Horne-Shimony-Holt (CHSH) Bell test. We show that this threshold can be lowered by using multiple copies of the two-qubit maximally entangled state. We get the upper bounds 80.86%80.86\%80.86%, 73.99%73.99\%73.99% and 69.29%69.29\%69.29% on the symmetric detection efficiency threshold for two, three and four copies of the state, where the respective number of measurements per party are 4, 8 and 16. However, in the case of four copies the result is partly due to a heuristic method. In order to get the corresponding Bell inequalities we made use of linear programming for two copies of the state and convex optimization based on Gilbert algorithm for three and four copies of the state.
In this paper, we investigate the critical efficiency of detectors to observe Bell nonlocality using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles, such as hyperentangled states, and the product of Pauli measurements. It is known that in a Clauser-Horne-Shimony-Holt (CHSH) Bell test the symmetric detection efficiency of 82.8482.84%82.84 can be tolerated for the two-qubit maximally entangled state. We beat this enigmatic threshold by entangling two particles with multiple degrees of freedom. The obtained upper bounds of the symmetric detection efficiency thresholds are 80.8680.86%80.86, 73.9973.99%73.99, and 69.2969.29%69.29 for two, three, and four copies of the two-qubit maximally entangled state, respectively. The number of measurements and outcomes in the respective cases are 4, 8, and 16. To find the improved thresholds, we use large-scale convex optimization tools, which allows us to significantly go beyond state-of-the-art results. The proof is exact up to three copies, while for four copies it is due to reliable numerical computations. Specifically, we used linear programming to obtain the two-copy threshold and the corresponding Bell inequality, and convex optimization based on Gilbert's algorithm for three and four copies of the two-qubit state. We show analytically that the symmetric detection efficiency threshold decays exponentially with the number of copies of the two-qubit state. Our techniques can also be applied to more general Bell nonlocality scenarios with more than two parties.
Recently, V\'ertesi and Bene [Phys. Rev. A 82, 062115 (2010)] derived a two-qubit Bell inequality ICH3{I}_{\mathrm{CH}3}ICH3​, which they show to be maximally violated only when more general positive operator-valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element POVM involved in the Bell test and obtain a higher quantum bound for the ICH3{I}_{\mathrm{CH}3}ICH3​ inequality. With a higher quantum bound for ICH3{I}_{\mathrm{CH}3}ICH3​, we investigate whether there is an experimental setup that can be used for observing that POVMs give higher violations in Bell tests based on this inequality. We analyze the maximum errors supported by the inequality to identify a source of entangled photons that can be used for the test. Then we study whether POVMs are also relevant in the more realistic case where partially entangled states are used in the experiment. Finally, we investigate the required efficiencies of the ICH3{I}_{\mathrm{CH}3}ICH3​ inequality, and the type of measurements involved, for closing the detection loophole. We obtain the result that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal required detection efficiency (in the case of two qubits) of the Clauser-Horne-Bell inequality.
Abstract Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.