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Physica Scripta

The iteration formula of (n, 2, d) full-correlated multi-component Bell function and its applications

Hui-Xian Meng, Yu Zhang, Xing-Yan Fan, Jie Zhou, Wei-Min Shang, Jing‐Ling Chen

Published Jul 25, 2024• 0 citations
Publisher

Abstract

Abstract No matter for a scientific or technological reason, constructing Bell inequalities for multi-partite and high-dimensional systems is a significant task. The Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) inequality is expressed in the form of iteration formula and correlation functions, and is the generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multi-partite cases. In the sense of detecting the nonlocality of the noisy maximally entangled states with the most white noises for the corresponding quantum system, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality, expressed in the joint probabilities form, is the high-dimensional analogue of the Clauser-Horne (CH) inequalities on joint probabilities. In the sense of detecting the nonlocality of the noisy Greenberger-Horne-Zeilinger(GHZ) states with the most white noises for the n-qudit system, it is a challenging task to construct the multi-partite analogue of the CGLMP inequality with iteration formula and correlation functions. In this paper, we generalize the multi-component correlation functions [Phys. Rev. A 71, 032 107 (2005)] for bipartite d-dimensional systems to n-partite d-dimensional ones, introduce the general full-correlated multi-component Bell function I n , d , and construct the corresponding Bell inequality. By this way, we can reproduce both the CGLMP inequality and the Bell inequality in [Phys. Rev. A 71, 032 107 (2005)] for the case n = 2, and the MABK inequality for the case d = 2. Inspired by the iteration formula form of the MABK inequality, we prove that for prime d the general Bell function I n , d can be reformulated by iterating two Bell functions I n − 1 , d . As applications, for prime d, confined to the unbiased symmetric (d × 2)-port beam splitters and the noisy n-qudit GHZ states, we recover the most robust Bell inequalities for small n and d, such as for the (3, 2, 3), (4, 2, 3), (5, 2, 3), and (3, 2, 5) Bell scenarios, with the iteration formula and the most robust Bell inequalities for the (2, 2, d) scenario. This implies that the iteration formula is an efficient way of constructing the multi-partite analogues of the CGLMP inequality with correlation functions. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.

DOI
10.1088/1402-4896/ad67b2

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Abstract No matter for a scientific or technological reason, constructing Bell inequalities for multi-partite and high-dimensional systems is a significant task. The Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) inequality is expressed in the form of iteration formula and correlation functions, and is the generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multi-partite cases. In the sense of detecting the nonlocality of the noisy maximally entangled states with the most white noises for the corresponding quantum system, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality, expressed in the joint probabilities form, is the high-dimensional analogue of the Clauser-Horne (CH) inequalities on joint probabilities. In the sense of detecting the nonlocality of the noisy Greenberger-Horne-Zeilinger(GHZ) states with the most white noises for the n-qudit system, it is a challenging task to construct the multi-partite analogue of the CGLMP inequality with iteration formula and correlation functions. In this paper, we generalize the multi-component correlation functions [Phys. Rev. A 71, 032 107 (2005)] for bipartite d-dimensional systems to n-partite d-dimensional ones, introduce the general full-correlated multi-component Bell function I n , d , and construct the corresponding Bell inequality. By this way, we can reproduce both the CGLMP inequality and the Bell inequality in [Phys. Rev. A 71, 032 107 (2005)] for the case n = 2, and the MABK inequality for the case d = 2. Inspired by the iteration formula form of the MABK inequality, we prove that for prime d the general Bell function I n , d can be reformulated by iterating two Bell functions I n − 1 , d . As applications, for prime d, confined to the unbiased symmetric (d × 2)-port beam splitters and the noisy n-qudit GHZ states, we recover the most robust Bell inequalities for small n and d, such as for the (3, 2, 3), (4, 2, 3), (5, 2, 3), and (3, 2, 5) Bell scenarios, with the iteration formula and the most robust Bell inequalities for the (2, 2, d) scenario. This implies that the iteration formula is an efficient way of constructing the multi-partite analogues of the CGLMP inequality with correlation functions. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.
It is very difficult and important to construct Bell inequalities for n-partite, k-settings of measurement, and d-dimensional (n,k,d) systems. Inspired by the iteration formula form of the Mermin-Ardehali-Belinski{\uı}-Klyshko (MABK) inequality, we generalize the multi-component correlation functions for bipartite d-dimensional systems to n-partite ones, and construct the corresponding Bell inequality. The Collins-Gisin-Linden-Massar-Popescu inequality can be reproduced by this way. The most important result is that for prime d the general Bell function in full-correlated multi-component correlation function form for (n,2,d) systems can be reformulated in iteration formula by two full-correlated multi-component Bell functions for (n-1,2,d) systems. As applications, we recover the MABK inequality and the most robust coincidence Bell inequalities for (3,2,3),(4,2,3),(5,2,3), and (3,2,5) Bell scenarios with this iteration formula. This implies that the iteration formula is an efficient way of constructing multi-partite Bell inequalities. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.
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.
Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities''. Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3N=3N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.
Self-testing is a device-independent technique that allows for verification of unknown quantum systems only when observed correlations achieve some quantum supremum. In this paper, we present self-testing schemes for a class of multipartite Bell inequalities with a specific structure, which we refer to as the unit. Our research is based on the graph-theoretic approach, where correlations are characterized via Lov\'asz theta body in the framework of exclusivity graphs. The derived statements are subsequently applied to prove the self-testability of NNN-partite Mermin-Ardehali-Belinski\ifmmode \check{i}\else \v{i}\fi{}-Klyshko (MABK) family of Bell inequalities. In particular, we take special cases to demonstrate that the maximal violation of the multipartite MABK inequality is a self-test for the corresponding multipartite Greenberger-Horne-Zeilinger (GHZ) state. Moreover, we propose a device-independent verification algorithm for distributed blind quantum computation in the noisy intermediate-scale quantum era combined with multipartite GHZ state self-testing, where multiple quantum servers are involved and work in collaboration to reduce the burden on a single server.
The general form of the coincidence Bell inequalities for an arbitrary (n,2,3)(n,2,3)(n,2,3) scenario [i.e., an nnn-party, two-setting, and three-dimensional system (qutrit) scenario] is presented. To detect the nonlocal properties of the noisy nnn-qutrit Greenberger-Horne-Zeilinger states, in this work we investigate the most robust (4,2,3)-scenario and (5,2,3)-scenario coincidence Bell inequalities. By deriving the most robust $(n\ensuremath{-}1)$-party coincidence inequalities, we have established two tight and inequivalent (n,2,3)(n,2,3)(n,2,3) coincidence Bell inequalities with a visibility of 0.5 for n=4n=4n=4 and, similarly, two tight and inequivalent (n,2,3)(n,2,3)(n,2,3) coincidence Bell inequalities with a visibility of 0.488785 for n=5n=5n=5. These inequalities are all tight. To our knowledge, up to now these inequalities have been the most robust Bell inequalities for the corresponding scenarios. Our results are useful for building the iteration formula of (n,2,3)(n,2,3)(n,2,3)-scenario coincidence Bell inequalities with the lowest critical visibility.