Creating large-scale entanglement lies at the heart of many quantum information processing protocols and the investigation of fundamental physics. For multipartite quantum systems, it is crucial to identify not only the presence of entanglement but also its detailed structure. This is because in a generic experimental situation with sufficiently many subsystems involved, the production of so-called genuine multipartite entanglement remains a formidable challenge. Consequently, focusing exclusively on the identification of this strongest type of entanglement may result in an all or nothing situation where some inherently quantum aspects of the resource are overlooked. On the contrary, even if the system is not genuinely multipartite entangled, there may still be many-body entanglement present in the system. An identification of the entanglement structure may thus provide us with a hint about where imperfections in the setup may occur, as well as where we can identify groups of subsystems that can still exhibit strong quantum-informationprocessing capabilities. However, there is no known efficient methods to identify the underlying entanglement structure. Here, we propose two complementary families of witnesses for the identification of such structures. They are based, respectively, on the detection of entanglement intactness and entanglement depth, each applicable to an arbitrary number of subsystems and whose evaluation requires only the implementation of solely two local measurements. Our method is also robust against noises and other imperfections, as reflected by our experimental implementation of these tools to verify the entanglement structure of five different eight-photon entangled states. In particular, we demonstrate how their entanglement structure can be precisely and systematically inferred from the experimental measurement of these witnesses. In achieving this goal, we also illustrate how the same set of data can be classically postprocessed to learn the most about the measured system.

In quantum information, lifting is a systematic procedure that can be used to derive-when provided with a seed Bell inequality-other Bell inequalities applicable in more complicated Bell scenarios. It is known that the procedure of lifting introduced by Pironio [J. Math. Phys. A 46, 062112 (2005)] preserves the facet-defining property of a Bell inequality. Lifted Bell inequalities therefore represent a broad class of Bell inequalities that can be found in all Bell scenarios. Here, we show that the maximal value of any lifted Bell inequality is preserved for both the set of nonsignaling correlations and quantum correlations. Despite the degeneracy in the maximizers of such inequalities, we show that the ability to self-test a quantum state is preserved under these lifting operations. In addition, except for outcome-lifting, local measurements that are self-testable using the original Bell inequality-despite the degeneracy-can also be self-tested using any lifted Bell inequality derived therefrom. While it is not possible to self-test all the positive-operator-valued measure elements using an outcome-lifted Bell inequality, we show that partial, but robust self-testing statements on the underlying measurements can nonetheless be made from the quantum violation of these lifted inequalities. We also highlight the implication of our observations on the usefulness of using lifted Bell-like inequalities as a device-independent witnesses for entanglement depth. The impact of the aforementioned degeneracy on the geometry of the quantum set of correlations is briefly discussed.

Techniques developed for device-independent characterizations allow one to certify certain physical properties of quantum systems without assuming any knowledge of their internal workings. Such a certification, however, often relies on the employment of device-independent witnesses catered for the particular property of interest. In this work, we consider a one-parameter family of multipartite, two-setting, two-outcome Bell inequalities and demonstrate the extent to which they are suited for the device-independent certification of genuine many-body entanglement (and hence the entanglement depth) present in certain well-known multipartite quantum states, including the generalized Greenberger-Horne-Zeilinger (GHZ) states with unbalanced weights, the higher-dimensional generalizations of balanced GHZ states, and the W states. As a by-product of our investigations, we have found that, in contrast with well-established results, provided trivial qubit measurements are allowed, full-correlation Bell inequalities can also be used to demonstrate the nonlocality of weakly entangled unbalanced-weight GHZ states. Besides, we also demonstrate how two-setting, two-outcome Bell inequalities can be constructed, based on the so-called GHZ paradox, to witness the entanglement depth of various graph states, including the ring graph states, the fully connected graph states, and some linear graph states, etc.