ABSTRACT Quantum tests directly reveal the incompatibility of quantum and classical predictions. The violation of such a test establishes a quantifiable gap between the capabilities of observers with access to classical resources and those utilizing quantum entanglement. Typically, larger violations indicate a higher degree of quantumness—for instance, enabling quantum self‐testing in Bell scenarios, where achieving maximal violation is highly desirable. The self‐guided method, initially developed for finding maximal Bell violations, is later extended to complex variables. Here, we apply and compare both variants to determine the maximal violation of the instrumental inequality. The complex‐number variant generally exhibits faster convergence, except during the initial phase. This improvement is more evident in instrumental tests, likely due to their simpler causal structure compared to Bell scenarios. Our findings establish the self‐guided method with complex variables as a generally applicable and competitive approach for robustly and efficiently identifying maximal violations across diverse quantum tests.
ABSTRACT Quantum tests directly reveal the incompatibility of quantum and classical predictions. The violation of such a test establishes a quantifiable gap between the capabilities of observers with access to classical resources and those utilizing quantum entanglement. Typically, larger violations indicate a higher degree of quantumness—for instance, enabling quantum self‐testing in Bell scenarios, where achieving maximal violation is highly desirable. The self‐guided method, initially developed for finding maximal Bell violations, is later extended to complex variables. Here, we apply and compare both variants to determine the maximal violation of the instrumental inequality. The complex‐number variant generally exhibits faster convergence, except during the initial phase. This improvement is more evident in instrumental tests, likely due to their simpler causal structure compared to Bell scenarios. Our findings establish the self‐guided method with complex variables as a generally applicable and competitive approach for robustly and efficiently identifying maximal violations across diverse quantum tests.
In recent decades, a great variety of research and applications concerning Bell nonlocality have been developed with the advent of quantum information science. Providing that Bell nonlocality can be revealed by the violation of a family of Bell inequalities, finding maximal Bell violation (MBV) for unknown quantum states becomes an important and inevitable task during Bell experiments. In this paper we introduce a self-guided method to find MBVs for unknown states using a stochastic gradient ascent algorithm (SGA), by parametrizing the corresponding Bell operators. For three investigated systems (two qubit, three qubit, and two qutrit), this method can ascertain the MBV of general two-setting inequalities within 100 iterations. Furthermore, we prove SGA is also feasible when facing more complex Bell scenarios, e.g., d-setting d-outcome Bell inequality. Moreover, compared to other possible methods, SGA exhibits significant superiority in efficiency, robustness, and versatility.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
An investigated process can be studied in terms of the causal relations among the involved variables, representing it as a causal model. Some causal models are particularly relevant, since they can be tested through mathematical constraints between the joint probability distributions of the observables. This is a valuable tool because, if some data violates the constraints of a causal model, the implication is that the observed statistics is not compatible with that causal structure. Strikingly, when non-classical correlations come to play, a discrepancy between classical and quantum causal predictions can arise, producing a quantum violation of the classical causal constraints. The simplestscenario admitting such quantum violation is given by the instrumental causal processes. Here, we experimentally violate an instrumental test on a photonic platform and show how the quantum correlations violating the CHSH inequality can be mapped into correlations violating an instrumental test, despite the different forms of non-locality they display. Indeed, starting from a Bell-like scenario, we recover the violation of the instrumental scenario through a map between the two behaviours, which includes a post-selection of data and then we test an alternative way to violate the CHSH inequality, adopting the instrumental process platform.
Self-testing protocols enable the certification of quantum devices without demanding full knowledge about their inner workings. A typical approach in designing such protocols is based on observing nonlocal correlations which exhibit maximum violation in a Bell test. We show that in the Bell experiment known as Hardy's test of nonlocality, not only does the maximally nonlocal correlation self-test a quantum state, rather a nonmaximal nonlocal behavior can serve the same purpose. We, in fact, completely characterize all such behaviors leading to a self-test of every pure two-qubit entangled state except for the maximally entangled ones. Apart from presenting an alternative self-testing protocol, our method provides a powerful tool towards characterizing the complex boundary of the set of quantum correlations.
Since Bell's theorem, it is known that local realism fails to explain quantum phenomena. Bell inequality violations manifestly show the incompatibility of quantum theory with classical notions of cause and effect. As recently found, however, the instrumental scenario-a pivotal tool in causal inference-allows for nonclassicality signatures going beyond this paradigm. If we are not limited to observational data and can intervene in our setup, then we can witness quantum violations of classical bounds on the causal influence among the involved variables even when no Bell-like violation is possible. That is, through interventions, the quantum behavior of a system that would seem classical can be demonstrated. Using a photonic setup-faithfully implementing the instrumental causal structure and switching between observation and intervention run by run-we experimentally witness such a nonclassicality. We also test quantum bounds for the causal influence, showing that they provide a reliable tool for quantum causal modeling.