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

István Márton, Erika Bene, Tamás Vértesi

Published Mar 18, 2021• 1 citation
OpenAlex

Abstract

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.

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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

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

Analyzing asymmetric nonlocality experiments with relaxed conditions

Daniel Dilley

It is already known that one can always find a set of measurements on any two-qubit entangled state that will lead to a violation of the CHSH inequality. We provide an explicit state in terms of the angle between Alice's choice of measurements and the angle between Bob's choice of measurements, such that the CHSH inequality is always violated provided Alice's or Bob's choice of inputs are not collinear. We prove that inequalities with a corresponding Bell operator written as a linear combination of tensor products of Pauli matrices, excluding the identity, will generate the most nonlocal correlations using maximally entangled states in our experiment. From this result and a proposition from Horodecki et. al., we are able to construct the state that generates these optimal correlations. To achieve this state in a lab, one party must rotate their qubit using the orthogonal operation we provide and also rotate their Bloch sphere such that all their measurements lie in the same plane. We provide a comprehensive study of how Bell inequalities change when experiments introduce error via imperfect detection efficiency. The original cases of perfect efficiency are covered first and then a more realistic approach, when inefficient detectors are used, will follow. It is shown that less entanglement is needed to demonstrate more nonlocality in some Clasure-Horne-Shimony-Holt (CHSH) experiments when detector inefficiency is introduced. An example of this is shown for any given specific set of measurements in the CHSH Bell experiment. This occurs when one party has a detector of efficiency for each choice of input and the other party makes projective measurements. The efficiency can be pushed down to fifty percent while still violating the CHSH inequality, and for the experimental set-up illustrated, there is more nonlocality with less entanglement. Furthermore, it is shown that if the first party has an imperfect detector for only one choice of inputs rather than two, the efficiency can be brought down arbitrarily close to zero percent while still violating the CHSH inequality. Historically, nonlocality and entanglement were viewed as two equivalent resources, but recently this equality has come under question; these results further support this fundamental difference. Further more, we introduce Mermin's game in the case of relaxed conditions. The original constraints were that when the detectors in separate labs of a two-qubit experiment are in the same setting, then the results should be the same. We require that the outcomes are the same at least part of the time, given by some epsilon variable. Initially, one could find a maximum violation of one-fourth by allowing to parties to share the singlet state and have measurement settings one-hundred and twenty degrees apart from one another. By allowing some epsilon error in the perfect correlations regime, one can find a maximum violation of minus one plus the square root of two using the singlet state and measurement inputs that achieve Tsirelson's bound for the CHSH experiment. The reason is that we show Mermin's inequality is technically the CHSH inequality "in disguise", but with using constraints the CHSH experiment does not use. We derive Mermin's inequality under new conditions and give the projective measurements needed to violate maximally.

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Detecting Bell-CHSH nonlocality of pure symmetric three-qubit states

K. Anjali, Akshata Shenoy H., H. S. Karthik +3 more

We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We show that the reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state, do not violate the CHSH inequality and hence they are CHSH-local . However, when Alice and Bob perform the CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. It is also shown that two different classes of three-qubit pure symmetric states, consisting of two and three distinct constituent spinors (qubits) respectively, can be distinguished through the conditional CHSH non-locality test.

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How likely are you to observe non-locality with imperfect detection efficiency and random measurement settings?

Paweł Cieśliński, Tamás Vértesi, Mateusz Kowalczyk +1 more

Abstract Imperfect detection efficiency remains one of the major obstacles in achieving loophole-free Bell tests over long distances. At the same time, the challenge of establishing a common reference frame for measurements becomes more pronounced as the separation between parties increases. In this work, we tackle both of these issues by examining the impact of limited detection efficiency on the probability of Bell inequality violation with Haar random measurement settings. We derive analytical lower bounds on the violation probability for a two-qubit maximally entangled state, which is tight for correlation inequalities and perfect detection efficiencies. We further investigate it numerically for more qubits and settings using two detection efficiency models and an original method based on the linear programming. Beyond that, we show that the so-called typicality of Bell inequality violation, i.e. almost certain violation of local realism with sufficiently many particles or random measurement directions, holds even if the detection efficiency is limited. Our findings reveal that increasing the number of measurement settings can compensate for efficiency losses above the critical threshold. Additionally, we determine critical detection efficiencies for both two-party and three-party scenarios. For two parties, we recover previously established results, while for three parties, we derive a symmetric critical efficiency of η crit = 2 / 3 for the W and GHZ states within the binning model. In cases involving the no-detection outcome, we present a modified inequality using a pair of orthogonal observables for each party with η crit ≈ 0.7208 , which is notably less than 0.75 for the GHZ state. These results offer deeper insights into the limitations and possibilities for certifying non-locality in the presence of limited detection efficiency, shedding light on its robustness in practical Bell tests.

arXiv:2503.21519•New Journal of Physics•Jun 27, 2025•1 citation

Self-testing two-qubit maximally entangled states from generalized Clauser-Horne-Shimony-Holt tests

Xavier Valcarce, Julian Zivy, Nicolas Sangouard +1 more

Device-independent certification, also known as self-testing, aims at guaranteeing the proper functioning of untrusted and uncharacterized devices. For example, the quality of an unknown source expected to produce two-qubit maximally entangled states can be evaluated in a bipartite scenario, each party using two binary measurements. The most robust approach consists in deducing the fidelity of produced states with respect to a two-qubit maximally entangled state from the violation of the CHSH inequality. In this paper, we show how the self-testing of two-qubit maximally entangled states is improved by a refined analysis of measurement statistics. The use of suitably chosen Bell tests, depending on the observed correlations, allows one to conclude higher fidelities than ones previously known. In particular, nontrivial self-testing statements can be obtained from correlations that cannot be exploited by a CHSH-based self-testing strategy. Our results not only provide insight into the set of quantum correlations suited for self-testing, but also facilitate the experimental implementations of device-independent certifications.

arXiv:2011.03047•Physical Review Research•Jan 24, 2022•9 citations
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.
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.
It is already known that one can always find a set of measurements on any two-qubit entangled state that will lead to a violation of the CHSH inequality. We provide an explicit state in terms of the angle between Alice's choice of measurements and the angle between Bob's choice of measurements, such that the CHSH inequality is always violated provided Alice's or Bob's choice of inputs are not collinear. We prove that inequalities with a corresponding Bell operator written as a linear combination of tensor products of Pauli matrices, excluding the identity, will generate the most nonlocal correlations using maximally entangled states in our experiment. From this result and a proposition from Horodecki et. al., we are able to construct the state that generates these optimal correlations. To achieve this state in a lab, one party must rotate their qubit using the orthogonal operation we provide and also rotate their Bloch sphere such that all their measurements lie in the same plane. We provide a comprehensive study of how Bell inequalities change when experiments introduce error via imperfect detection efficiency. The original cases of perfect efficiency are covered first and then a more realistic approach, when inefficient detectors are used, will follow. It is shown that less entanglement is needed to demonstrate more nonlocality in some Clasure-Horne-Shimony-Holt (CHSH) experiments when detector inefficiency is introduced. An example of this is shown for any given specific set of measurements in the CHSH Bell experiment. This occurs when one party has a detector of efficiency for each choice of input and the other party makes projective measurements. The efficiency can be pushed down to fifty percent while still violating the CHSH inequality, and for the experimental set-up illustrated, there is more nonlocality with less entanglement. Furthermore, it is shown that if the first party has an imperfect detector for only one choice of inputs rather than two, the efficiency can be brought down arbitrarily close to zero percent while still violating the CHSH inequality. Historically, nonlocality and entanglement were viewed as two equivalent resources, but recently this equality has come under question; these results further support this fundamental difference. Further more, we introduce Mermin's game in the case of relaxed conditions. The original constraints were that when the detectors in separate labs of a two-qubit experiment are in the same setting, then the results should be the same. We require that the outcomes are the same at least part of the time, given by some epsilon variable. Initially, one could find a maximum violation of one-fourth by allowing to parties to share the singlet state and have measurement settings one-hundred and twenty degrees apart from one another. By allowing some epsilon error in the perfect correlations regime, one can find a maximum violation of minus one plus the square root of two using the singlet state and measurement inputs that achieve Tsirelson's bound for the CHSH experiment. The reason is that we show Mermin's inequality is technically the CHSH inequality "in disguise", but with using constraints the CHSH experiment does not use. We derive Mermin's inequality under new conditions and give the projective measurements needed to violate maximally.
We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We show that the reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state, do not violate the CHSH inequality and hence they are CHSH-local . However, when Alice and Bob perform the CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. It is also shown that two different classes of three-qubit pure symmetric states, consisting of two and three distinct constituent spinors (qubits) respectively, can be distinguished through the conditional CHSH non-locality test.
Abstract Imperfect detection efficiency remains one of the major obstacles in achieving loophole-free Bell tests over long distances. At the same time, the challenge of establishing a common reference frame for measurements becomes more pronounced as the separation between parties increases. In this work, we tackle both of these issues by examining the impact of limited detection efficiency on the probability of Bell inequality violation with Haar random measurement settings. We derive analytical lower bounds on the violation probability for a two-qubit maximally entangled state, which is tight for correlation inequalities and perfect detection efficiencies. We further investigate it numerically for more qubits and settings using two detection efficiency models and an original method based on the linear programming. Beyond that, we show that the so-called typicality of Bell inequality violation, i.e. almost certain violation of local realism with sufficiently many particles or random measurement directions, holds even if the detection efficiency is limited. Our findings reveal that increasing the number of measurement settings can compensate for efficiency losses above the critical threshold. Additionally, we determine critical detection efficiencies for both two-party and three-party scenarios. For two parties, we recover previously established results, while for three parties, we derive a symmetric critical efficiency of η crit = 2 / 3 for the W and GHZ states within the binning model. In cases involving the no-detection outcome, we present a modified inequality using a pair of orthogonal observables for each party with η crit ≈ 0.7208 , which is notably less than 0.75 for the GHZ state. These results offer deeper insights into the limitations and possibilities for certifying non-locality in the presence of limited detection efficiency, shedding light on its robustness in practical Bell tests.
Device-independent certification, also known as self-testing, aims at guaranteeing the proper functioning of untrusted and uncharacterized devices. For example, the quality of an unknown source expected to produce two-qubit maximally entangled states can be evaluated in a bipartite scenario, each party using two binary measurements. The most robust approach consists in deducing the fidelity of produced states with respect to a two-qubit maximally entangled state from the violation of the CHSH inequality. In this paper, we show how the self-testing of two-qubit maximally entangled states is improved by a refined analysis of measurement statistics. The use of suitably chosen Bell tests, depending on the observed correlations, allows one to conclude higher fidelities than ones previously known. In particular, nontrivial self-testing statements can be obtained from correlations that cannot be exploited by a CHSH-based self-testing strategy. Our results not only provide insight into the set of quantum correlations suited for self-testing, but also facilitate the experimental implementations of device-independent certifications.