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arXiv:2508.03525 · Physical Review A

Entanglement detection beyond the local bound with coarsely calibrated measurements

Liang-Liang Sun, Yongshun Song, Sixia Yu

Published Jan 6, 2026• 0 citations
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Abstract

Bell's test, initially devised to distinguish quantum theory from local hidden variable models through {violations of local bounds}, is also a common tool for detecting entanglement. For this purpose, one can assume the quantum description of devices and use available information to strengthen the bound for separable states, which may go beyond the local bound, enabling more efficient entanglement detection. Here we present a systematic approach for strengthening Bell inequalities for qubit systems, using Mermin-Klyshko-Bell inequalities as examples, by considering measurement devices that are coarsely calibrated only by their ability to generate nonlocal correlations without requiring precise quantum characterization. In the case of bipartite and tripartite systems, we derive trade-offs between upper bounds for separable states and general states in terms of structure functions to optimize the entanglement detection. We then strengthen $n$-partite Bell inequalities for the detection of states exhibiting a diversity of entanglement structures such as genuine multipartite entanglement. For general Bell scenarios with some measurements characterized, we demonstrate that entanglement can also be detected with some local correlations by exploiting the Navascués-Pironio-Acín hierarchy of tests.

DOI
10.1103/msmm-46n3

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Bell inequality tests where the detection efficiency is below a certain threshold ηcrit can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low ηcrit and relatively low dimension d of the local quantum systems. The method has two steps. First, we show a family of bipartite Bell inequalities for which, for correlations produced by maximally entangled states, ηcrit can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small ηcrit . We present examples in which, for maximally entangled states, ηcrit≤0.516 for d=16 , ηcrit≤0.407 for d=28 , and ηcrit≤0.326 for d=32 . We also show evidence that the upper bound for ηcrit can be lowered down to 0.415 for d=16 and present a method to make the upper bound of ηcrit arbitrarily small by increasing the dimension and the number of settings. All these upper bounds for ηcrit are valid (as it is the case in the literature) assuming no noise. The second step is based on the observation that, using the initial state and measurement settings identified in the first step, we can construct Bell inequalities with smaller ηcrit and better noise robustness. For that, we use a modified version of Gilbert's algorithm that takes advantage of the automorphisms of the graphs used in the first step. We illustrate its power by explicitly developing an example in which ηcrit is 12.38% lower and the required visibility is 14.62% lower than the upper bounds obtained in the first step. The tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.
No-signaling theories, which can contain nonlocal correlations stronger than quantum correlations but limited by the no-signaling condition, have deepened our understanding of the quantum theory. In practice, Bell tests are powerful tools to certify nonlocality, but their effectiveness is limited by the detector efficiency. In this work, we provide almost tight detection efficiency bounds for showing the nonlocality of no-signaling theories, by introducing a general class of Bell inequalities. In particular, we provide a tight bound for the bipartite case and an asymptotic tight bound for the multipartite case. The tightness of these bounds shows that they well characterize the structure of no-signaling theories. The bounds also imply that the detector efficiency requirement can be made arbitrarily low in both bipartite and multipartite cases by increasing the number of measurement settings. Furthermore, our work sheds light on the detector efficiency requirement for showing the nonlocality of the quantum theory.
Bell inequalities have traditionally been used to demonstrate that quantum theory is nonlocal, in the sense that there exist correlations generated from composite quantum states that cannot be explained by means of local hidden variables. With the advent of device-independent quantum information protocols, Bell inequalities have gained an additional role as certificates of relevant quantum properties. In this work, we consider the problem of designing Bell inequalities that are tailored to detect maximally entangled states. We introduce a class of Bell inequalities valid for an arbitrary number of measurements and results, derive analytically their tight classical, nonsignaling, and quantum bounds and prove that the latter is attained by maximally entangled states. Our inequalities can therefore find an application in device-independent protocols requiring maximally entangled states.
Bell inequalities have traditionally been used to demonstrate that quantum theory is nonlocal, in the sense that there exist correlations generated from composite quantum states that cannot be explained by means of local hidden variables. With the advent of device-independent quantum information processing, Bell inequalities have gained an additional role as certificates of relevant quantum properties. In this work we consider the problem of designing Bell inequalities that are tailored to detect the presence of maximally entangled states. We introduce a class of Bell inequalities valid for an arbitrary number of measurements and results, derive analytically their maximal violation and prove that it is attained by maximally entangled states. Our inequalities can therefore find an application in device-independent protocols requiring maximally entangled states.
Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser–Horne–Shimony–Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k -way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations—the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.
Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.