Abstract
The Bell test, as an important method for detecting nonlocality, is widely used in device-independent quantum information processing tasks. The security of these tasks is based on an assumption called measurement independence. Since this assumption is difficult to be guaranteed in practical Bell tests, it is meaningful to consider the effect of reduced measurement independence (i.e., measurement dependence) on Bell tests. Some research studies have shown that nonlocality can be detected even if measurement dependence exists. However, the relevant results are all based on bipartite Bell tests, and the results for multipartite Bell tests are still missing. In this paper, we explore this problem in the tripartite Svetlichny test. By considering flexible lower and upper bounds on the degree of measurement dependence, we obtain the relation among measurement dependence, guessing probability, and the maximal value of Svetlichny inequality. Our results reveal the case in which genuine nonlocality is nonexistent; at this point, the outcomes of the Bell test cannot be applied in device-independent quantum information processing tasks.
1 Introduction
Quantum nonlocality is a critical resource in device-independent quantum information processing tasks such as quantum key distribution [1–3], random number generation [4–7], self-testing [8,9], and private query [10,11]. The phenomenon of quantum nonlocality, first brought to light in the famous debate between Einstein, Podolsky, and Rosen in 1935 [12], was later given a testable framework by Bell through his inequality theorem formulated in 1964 [13].
A bipartite Bell test involves two distant parties: Alice and Bob. Each party randomly selects measurement settings Ax(x ∈ {0, 1}) and By(y ∈ {0, 1}) and obtains outcomes a ∈ {0, 1} and b ∈ {0, 1}, respectively. After many rounds of experiments, the statistical correlations are characterized by the joint probability distribution p(a, b|Ax, By). Bell inequality can be defined by a linear combination of p(a, b|Ax, By):where ⟨AxBy⟩ = ∑a,b(−1)a+bp(a, b|Ax, By). This inequality is known as the CHSH inequality [14]. The statistical correlations produced by the classical system can reach the local upper bound of CHSH inequality of 2. In quantum mechanics, measurements acting on quantum entanglement states can violate the CHSH inequality and result in a bound value of up to .
Since the 1980s, Bell test experiments have been realized [15–18], but it has been found that these experiments suffer a loophole called randomness loophole. The Bell test requires that the selection of measurements is completely random. This is a basic assumption for the Bell test, called measurement independence. If measurement independence cannot be guaranteed in the experiment, the randomness loophole will be opened up. The adversary Eve is able to simulate quantum nonlocality with classical systems by possessing a priori knowledge of measurement settings, which threatens the security of device-independent quantum information processing tasks. Nevertheless, measurement independence is difficult to be guaranteed in practical Bell tests. Many researchers have attempted to explore how relaxing the measurement independence assumption (i.e., measurement dependence) would affect the Bell test [19–30].
In 2010, Hall [19] proposed the quantification of measurement dependence and constructed a local deterministic model to simulate singlet state correlations. In 2012, Koh et al. [21] explored the effects of measurement dependence on the CHSH–Bell test. By considering the relation among measurement dependence, guessing probability, and the maximal value of CHSH inequality, they bounded the capabilities of the adversary. It is worth mentioning that in [23,24], Pütz et al. improved the quantification of measurement dependence and developed a framework for measurement dependence locality. This notable approach was used by Yuan et al. [31] to consider the effects of measurement dependence on the CHSH–Bell test.
With the development of quantum information and quantum computing, more complex Bell test models deserve to be considered. Many research studies have been devoted to measurement dependence based on the Bell test with multiple measurements, multiple outcomes, or asymmetric Bell inequality [31–35]. While previous discussions on measurement dependence mainly focus on the bipartite Bell test, the multipartite Bell test still deserves to be explored. In this paper, we explore how measurement dependence affects the tripartite Svetlichny test. By introducing the quantification of measurement dependence in the tripartite Svetlichny test, the relation among measurement dependence, guessing probability, and the maximal value of Svetlichny inequality is obtained. The result demonstrates the capabilities of the adversary Eve to simulate quantum nonlocality using classical systems, which is crucial in device-independent quantum information processing tasks.
This paper is organized as follows: Section 2 provides a brief introduction of Svetlichny inequality and the tripartite Bell test. The main results are presented in Section 3. Section 4 analyzes the capabilities of the adversary when considering random number generation. The conclusion is presented in Section 5.
2 Preliminaries
In this section, some relevant preliminaries are given.
2.1 Local hidden variable model
Bell’s theorem states that the correlations produced by the quantum system cannot be explained by the local hidden variable (LHV) model. For the CHSH–Bell test, the statistical correlation p(a, b|Ax, By) admits the following decomposition:
Here, λ is the local hidden variable which denotes all the factors that may affect the outcomes.
Additional assumptions may lead to restrictions on Eq. 2. The first one is called local causality:
This assumption requires that the outcomes of each party are only dependent on the inputs of that party and the local hidden variable λ. In a practical Bell test, one guarantees the assumption by ensuring that the two devices are spatially separated. Local causality can also be viewed as the union of two assumptions: parameter independence and outcome independence.
The second assumption is called measurement independence:
Measurement independence requires that the selection of inputs of each party is independent of the local hidden variable λ. According to the Bayes theorem, Eq. 4 can also be written as
In a practical Bell test, one guarantees this assumption as much as possible using ideal randomness. After considering these two assumptions, the statistical correlation can be described as
If the statistical correlations can be written in the form of Eq. 6, they satisfy the CHSH inequality.
As a natural extension of the bipartite Bell test, the multipartite Bell test displays a more complex structure. We consider the simplest tripartite Bell test, which contains three distinct parties: Alice, Bob, and Charlie, whose devices are spatially separated from each other. Each of them has two inputs and two outcomes. The inputs are labeled as Aj, Bk, and Cl, where j, k, l ∈ {0, 1}, and the outcomes are labeled as a, b, c ∈ {0, 1}, respectively. These devices can be treated as black boxes, and an outcome will be given when an input of the devices is selected. After repeating this process several times, the statistical correlations p(a, b, c|Aj, Bk, Cl) are obtained. According to Bell’s theorem, a local correlation p(a, b, c|Aj, Bk, Cl) can be written aswhere λ is the local hidden variable and ∫dλp(λ) = 1.
We know that if Eq. 7 does not hold, then the correlations are nonlocal. However, several representations indicate that the correlations are nonlocal in the multipartite Bell test. For instance, if Alice is uncorrelated to Bob and Charlie in the tripartite case, then the correlations can be written in the following form:where ∫dλp(λ) = 1 if p(b, c|Bk, Cl, λ) is nonlocal. It is easy to find that if Eq. 8 violates the form of Eq. 7, the correlations are nonlocal. However, such correlations are strictly bipartite nonlocal, which is independent of the third party.
2.2 Svetlichny inequality
To distinguish the nonlocal correlations generated by all three parties, Svetlichny constructed an inequality in 1987, also known as Svetlichny inequality [36]. If the correlations produced by the tripartite Svetlichny test can be written in the formwhere ∫dλp(λ) + ∫dμp(μ) + ∫dνp(ν) = 1, then the correlations satisfy the following Svetlichny inequality:where ⟨AxByCz⟩ = ∑a,b,c(−1)a+b+cp(a, b, c|Ax, By, Cz). In quantum systems, the correlations produced by measurements that act on genuine tripartite entanglement states may violate the Svetlichny inequality, and the upper bound can be up to . Furthermore, the upper bound of Svetlichny inequality is up to 8 for the no-signaling theory.
2.3 Measurement dependence and guessing probability
As mentioned above, the assumption of local causality can be guaranteed by spatial separation. However, measurement independence is difficult to be realized in the practical Bell test. If the devices are potentially prepared by Eve, she can use local correlation to reproduce the quantum correlation by controlling the local hidden variable λ. Eve’s control on inputs is described by p(Aj, Bk, Cl|λ). If p(Aj, Bk, Cl|λ) = p(Aj, Bk, Cl), it means that the information Eve learned does not affect the inputs, also known as measurement independence. If p(Aj, Bk, Cl|λ) ≠p(Aj, Bk, Cl), this is called measurement dependence, which means that Eve can decide the inputs by controlling λ. The quantification is defined by the upper bound of conditional input probability distributions. In the tripartite Svetlichny test, the degree of measurement dependence is expressed aswhere . The case corresponds to measurement independence. The case Pup = 1 corresponds to complete measurement dependence, i.e., Eve has full control over at least one of the inputs via λ.
It has been shown that the lower bound of conditional input probability distributions is also an important parameter for quantifying measurement dependence [23]. This parameter characterizes the minimum randomness requirement of the inputs. A better quantification of measurement dependence can be obtained by combining the upper and lower bounds of conditional input probability distributions. In the tripartite Svetlichny test, it is expressed as follows:where and .
The guessing probability reflects the randomness. The adversary Eve tries to guess Alice’s outcomes: the more accurate Eve’s guess is, the less random the outcome will be. G(λ) denotes the marginal probability of Eve’s best guess for a given local hidden variable λ. In the tripartite Svetlichny test,
The guessing probability will then be given by
The maximal value G = 1 represents that Eve can guess all the outputs, which means that the outputs are completely deterministic. The minimal value represents that Eve cannot get extra information about the output via λ.
3 Results
In this section, the main results are given. According to the definitions given above, we obtain the relation between the value of the Svetlichny inequality with respect to the guessing probability and the degree of measurement dependence. Before we obtain the main result in this paper, we will first introduce the following lemma.
Lemma: The maximum possible value of Svetlichny inequality for the tripartite case , for any no-signaling model with (i.e., all inputs are equally likely), iswhere G and Pup are the guessing probability and degree of measurement dependence, respectively.
The degree of measurement dependence in this lemma is described as the upper bound of conditional input probability distributions p(Aj, Bk, Cl|λ). The proof of lemma is included in the proof of the theorem. According to the lemma and the definitions we introduced above, we obtain the relation with flexible lower and upper bounds of conditional input probability distributions.
Theorem: The maximum possible value of Svetlichny inequality for the tripartite case , for any no-signaling model with (i.e., all inputs are equally likely), is
In the following part, we will prove the theorem and the lemma together.
Proof: We start by defining the marginal probability aswhere j, k, l ∈ {0, 1}. Then, the remaining marginal probabilities containing one variable can be expressed as
According to the definition of the guessing probability, we get
Similarly, we define the marginal probability containing two variables aswhere j, k, l ∈ {0, 1}. Then, the remaining marginal probabilities containing two variables can be expressed as
We also define the joint probability p(0, 0, 0|Aj, Bk, Cl, λ) = fjkl. Then, all the remaining joint probabilities are expressed as follows:
Because of the positivity of probability, we obtain the range of fjkl:where djkl = 1 − mj − nk − ol + xjk + yjl + zkl. Since and , it can be extended to the general cases:
In order to simplify the representation, we will follow the techniques in [20]. By the equation in Appendix B of [20], we obtain
Similarly, the maximum case can also be extended to the following case:
Thus, fjkl satisfies
Based on the definition of ⟨AjBkCl⟩, we haveLet denote the expectation of the measurement outcomes for a fixed value of λ:Substituting the joint probability into Eq. 25, can be reproduced as
Thus, satisfies
The Svetlichny inequality for the tripartite case is described by
Combining the results obtained above, we can get
The specific processes of simplification are listed in the Supplementary Material.
In the following part, we consider the degree of measurement dependence Pup. Based on the definition of Pup, in the case of , we can always find that minj,k,lp(Aj, Bk, Cl|λ) = 0. In the case of , the minimum value is p(Aj, Bk, Cl|λ) = 1 − 7Pup. Then, we can conclude that
Now that we have completed the proof of the lemma, we will continue with the proof of the theorem.
According to the definition of the flexible upper bound and lower bound of measurement dependence, we havewhere j, k, l ∈ {0, 1}.
Let
It is easy to obtain the range of p(Aj, Bk, Cl|λ):
The normalization of p(Aj, Bk, Cl|λ) is proven as follows:where the last equality holds according to the normalization of p′(Aj, Bk, Cl|λ).
Thus, can be described by
According to Eq. 29 and Eq. 35, we obtain the relation between S3 and :
In the case of , i.e., 7Pup + Plow ≤ 1, we obtain
In the case of , i.e., 7Pup + Plow ≥ 1, we obtain
Consequently, based on the guessing probability G, the flexible bound of measurement dependence Pup and Plow, the Svetlichny inequality value can be described as
4 Discussion
In this section, the analysis of the adversary’s capability is given. Regarding the lemma, we describe it in Figure 1. For the case Pup = 1, the inputs of the devices are completely deterministic for the adversary Eve. She can construct a local strategy to preprogram the outcomes, and the value of Svetlichny inequality can be up to 8. This makes the outcomes seem random, but it is actually certain for Eve(G = 1). For the case , the inputs of the devices are completely random, and Eve has no a priori knowledge about the inputs. If she constructs a local strategy to preprogram the outcomes, the value of Svetlichny inequality will not exceed 4.
FIGURE 1
We also illustrate the theorem by presenting it in two cases. As described in Figure 2, for the case of or , the inputs are completely random, and the adversary Eve has no a priori knowledge of the inputs. In this case, if Eve constructs a deterministic strategy of outcomes, the Svetlichny inequality cannot be violated. Thus, the adversary Eve cannot fake true randomness. For the case of 7Pup + Plow > 1 or 7Pup + Plow < 1, Eve can obtain part of the information of inputs and then can construct a classical strategy to reach the maximum value of the Svetlichny inequality . In this case, we conclude that there is true randomness generation when .
FIGURE 2
5 Conclusion
In this paper, we explored the effect of measurement dependence on the tripartite Svetlichny test. Concretely, we showed the relation among measurement dependence, guessing probability, and the maximal violation of Svetlichny inequality that the adversary can fake. Using the degree of measurement dependence with flexible lower and upper bounds, we analyze the case that genuine tripartite nonlocality is nonexistent and give a security analysis of the device-independent quantum information processing tasks. Taking random number generation as an example, we considered the range that the adversary Eve can fake using the classical system. Attempts to the adversary with stronger ability merit further investigation. A natural extension of this work is to explore more different correlations, such as network nonlocality [37] and steering [38].
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.
Author contributions
R-ZL: writing–original draft, writing–review and editing, conceptualization, data curation, formal analysis, investigation, methodology, resources, software, supervision, validation, and visualization. D-DL: writing–original draft, writing–review and editing, conceptualization, data curation, formal analysis, and investigation. S-YW: data curation, formal analysis, methodology, project administration, software, visualization, and writing–review and editing. S-JQ: formal analysis, funding acquisition, methodology, project administration, supervision, and writing–review and editing. FG: formal analysis, funding acquisition, writing–original draft, writing–review and editing, and project administration. Q-YW: formal analysis, funding acquisition, project administration, resources, and writing–review and editing.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work is supported by the National Natural Science Foundation of China (Grant Nos 62272056, 62171056, and 62372048).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2024.1356682/full#supplementary-material
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Summary
Keywords
Bell test, Bell nonlocality, measurement dependence, Svetlichny inequality, device-independent quantum information task
Citation
Li R-Z, Li D-D, Wu S-Y, Qin S-J, Gao F and Wen Q-Y (2024) Tripartite Svetlichny test with measurement dependence. Front. Phys. 12:1356682. doi: 10.3389/fphy.2024.1356682
Received
16 December 2023
Accepted
25 January 2024
Published
08 February 2024
Volume
12 - 2024
Edited by
Xiao Yuan, Peking University, China
Reviewed by
Jinjing Shi, Central South University, China
He Lu, Shandong University, China
Xiaogang Li, Peking University, China
Updates
Copyright
© 2024 Li, Li, Wu, Qin, Gao and Wen.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Fei Gao, gaof@bupt.edu.cn; Su-Juan Qin, qsujuan@bupt.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.