Bell nonlocality describes a manifestation of quantum mechanics that cannot be explained by any local hidden variable model. Its origin lies in the nature of quantum entanglement, although understanding the precise relationship between nonlocality and entanglement has been a notorious open problem. In this paper, we develop a dynamical framework in which quantum Bell nonlocality emerges as a special form of entanglement and both are unified as resources under local operations and classical communication (LOCC). Our framework is built on the notion of classical and quantum processes, which are defined as channels that map elements between specific intervals in space and time. Entanglement is identified as a process that cannot be generated by LOCC while Bell nonlocality is the subset of these processes that have an instantaneous input-to-output delay time. LOCC preprocessing is a natural set of free operations in this theory, thereby providing previous nonlocality activation results a clear resource-theoretic foundation. We provide a systematic method to quantify the Bell nonlocality of a bipartite quantum channel. It is shown that both the relative entropy and the max relative entropy of nonlocality are nonadditive for a family of bipartite classical channels. This family includes the channel obtained when using the singlet state to maximally violate the CHSH inequality. We also find that the regularized relative entropy of Bell nonlocality provides an upper bound on the asymptotic rate of converting (i.e., simulating) many copies of one classical instantaneous resource to another.
Bell nonlocality describes a manifestation of quantum mechanics that cannot be explained by any local hidden variable model. Its origin lies in the nature of quantum entanglement, although understanding the precise relationship between nonlocality and entanglement has been a notorious open problem. In this paper, we develop a dynamical framework in which quantum Bell nonlocality emerges as a special form of entanglement and both are unified as resources under local operations and classical communication (LOCC). Our framework is built on the notion of classical and quantum processes, which are defined as channels that map elements between specific intervals in space and time. Entanglement is identified as a process that cannot be generated by LOCC while Bell nonlocality is the subset of these processes that have an instantaneous input-to-output delay time. LOCC preprocessing is a natural set of free operations in this theory, thereby providing previous nonlocality activation results a clear resource-theoretic foundation. We provide a systematic method to quantify the Bell nonlocality of a bipartite quantum channel. It is shown that both the relative entropy and the max relative entropy of nonlocality are nonadditive for a family of bipartite classical channels. This family includes the channel obtained when using the singlet state to maximally violate the CHSH inequality. We also find that the regularized relative entropy of Bell nonlocality provides an upper bound on the asymptotic rate of converting (i.e., simulating) many copies of one classical instantaneous resource to another.
Quantum mechanics is a highly nonlocal theory of nature. Quantum systems exhibit correlations which cannot be described by any classical theory of locality. We develop the resource theory of dynamical Bell nonlocality, which includes bipartite states, classical channels, quantum channels and measurements. In the state scenario, all separable states are Bell local. However, there exist mixed bipartite entangled states which also admit Bell local behaviour. To address this anomaly, we introduce the notion of fully Bell locality and show that all entangled states are Bell nonlocal, in the sense that they can be used to simulate at least one nonlocal bipartite Positive Operator Valued Measure (POVM) channel. We take a step further and generalise this result to bipartite entangled quantum channels. We then generalize the CHSH inequality from bipartite classical channels to bipartite POVM channels and devise a technique to check if a given bipartite POVM channel is nonlocal or not. Finally we provide a systematic method to quantify Bell nonlocality of bipartite quantum channels by extending any monotone for Bell nonlocality of classical channels to quantum channels and also introduce the precise definition of relative entropy of Bell nonlocality. We leave some open problems in the way.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.
A standard approach to quantifying resources is to determine which operations on the resources are freely available, and to deduce the partial order over resources that is induced by the relation of convertibility under the free operations. If the resource of interest is the nonclassicality of the correlations embodied in a quantum state, i.e., e n t a n g l e m e n t , then the common assumption is that the appropriate choice of free operations is Local Operations and Classical Communication (LOCC). We here advocate for the study of a different choice of free operations, namely, Local Operations and Shared Randomness (LOSR), and demonstrate its utility in understanding the interplay between the entanglement of states and the nonlocality of the correlations in Bell experiments. Specifically, we show that the LOSR paradigm (i) provides a resolution of the anomalies of nonlocality , wherein partially entangled states exhibit more nonlocality than maximally entangled states, (ii) entails new notions of genuine multipartite entanglement and nonlocality that are free of the pathological features of the conventional notions, and (iii) makes possible a resource-theoretic account of the self-testing of entangled states which generalizes and simplifies prior results. Along the way, we derive some fundamental results concerning the necessary and sufficient conditions for convertibility between pure entangled states under LOSR and highlight some of their consequences, such as the impossibility of catalysis for bipartite pure states. The resource-theoretic perspective also clarifies why it is neither surprising nor problematic that there are mixed entangled states which do not violate any Bell inequality. Our results motivate the study of LOSR-entanglement as a new branch of entanglement theory.
Bell nonlocality refers to correlations between two distant, entangled particles that challenge classical notions of local causality. Beyond its foundational significance, nonlocality is crucial for device-independent technologies like quantum key distribution and randomness generation. Nonlocality quickly deteriorates in the presence of noise, and restoring nonlocal correlations requires additional resources. These often come in the form of many instances of the input state and joint measurements, incurring a significant resource overhead. Here, we experimentally demonstrate that single copies of Bell-local states, incapable of violating any standard Bell inequality, can give rise to nonlocality after being embedded into a quantum network of multiple parties. We subject the initial entangled state to a quantum channel that broadcasts part of the state to two independent receivers and certify the nonlocality in the resulting network by violating a tailored Bell-like inequality. We obtain these results without making any assumptions about the prepared states, the quantum channel, or the validity of quantum theory. Our findings have fundamental implications for nonlocality and enable the practical use of nonlocal correlations in real-world applications, even in scenarios dominated by noise.
Locally classical behavior is often interpreted as evidence that a global classical description should exist. In particular, when individual processes admit classical (measure-and-prepare) realizations, it is natural to expect at least one compatible joint implementation that remains classical. We show that this intuition fails in a simple broadcast setting with one input system and two outputs. Specifically, we construct pairs of compatible channels that are entanglement-breaking individually and thus admit classical descriptions, yet every joint broadcast realization is necessarily Bell-nonlocal between the outputs, even for a maximally mixed input. This establishes a form of compatibility-induced activation of nonlocality: Bell nonlocality arises not from entanglement in the inputs or marginal channels, but solely from the requirement that these marginals admit a common global implementation. In this sense, compatibility is not merely a consistency condition on local descriptions, but a mechanism that can enforce nonclassical correlations at the global level.
Activation of Bell nonlocality refers to the phenomenon that some entangled mixed states that admit a local hidden variable model in the standard Bell scenario nevertheless reveal their nonlocal nature in more exotic measurement scenarios. We present such a scenario that involves broadcasting the local subsystems of a single-copy of a bipartite quantum state to multiple parties, and use the scenario to study the nonlocal properties of the two-qubit isotropic state: ρα=α|Φ+⟩⟨Φ+|+(1−α)14. We present two main results, considering that Nature allows for (i) the most general no-signalling correlations, and (ii) the most general quantum correlations at the level of any hidden variable theory. We show that the state does not admit a local hidden variable description for α>0.559 and α>12 , in cases (i) and (ii) respectively, which in both cases provides a device-independent certification of the entanglement of the state. These bounds are significantly lower than the previously best-known bound of 0.697 for both Bell nonlocality and device-independent entanglement certification using a single copy of the state. Our results show that strong examples of non-classicality are possible with a small number of resources.