Device-independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully ``causal'' simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables and the measurement settings, for the Clauser-Horne-Shimony-Holt (CHSH) scenario, and find that the maximal quantum violation requires a mutual information of just $\ensuremath{\sim}0.080$ bits. Any physical device built to implement this simulation allows an adversary to have full knowledge of a cryptographic key or ``random'' numbers generated by a device-independent protocol based on violation of the CHSH inequality. We also show that a previous model for the CHSH scenario, requiring only $\ensuremath{\sim}0.046$ bits to simulate the maximal quantum violation, corresponds to the most efficient ``retrocausal'' simulation, in which future measurement settings necessarily influence earlier source variables. This may be viewed either as an unphysical limitation of the prior model or as an argument for retrocausality on the grounds of its greater efficiency. Causal and retrocausal models are also discussed for maximally entangled two-qubit states, as well as superdeterministic, one-sided, and zigzag causal models.
Device-independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully causal'' simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables and the measurement settings, for the Clauser-Horne-Shimony-Holt (CHSH) scenario, and find that the maximal quantum violation requires a mutual information of just $\ensuremath{\sim}0.080$ bits. Any physical device built to implement this simulation allows an adversary to have full knowledge of a cryptographic key or random'' numbers generated by a device-independent protocol based on violation of the CHSH inequality. We also show that a previous model for the CHSH scenario, requiring only \ensuremath∼0.046 bits to simulate the maximal quantum violation, corresponds to the most efficient ``retrocausal'' simulation, in which future measurement settings necessarily influence earlier source variables. This may be viewed either as an unphysical limitation of the prior model or as an argument for retrocausality on the grounds of its greater efficiency. Causal and retrocausal models are also discussed for maximally entangled two-qubit states, as well as superdeterministic, one-sided, and zigzag causal models.
Quantum nonlocality stands as a resource for device independent quantum information processing (DIQIP), such as, for instance, device independent quantum key distribution. We investigate, experimentally, the assumption of limited measurement dependence, i.e., that the measurement settings used in Bell inequality tests or DIQIP are partially influenced by the source of entangled particle and/or by an adversary. Using a recently derived Bell-like inequality [G. Pütz, Phys. Rev. Lett. 113, 190402 (2014)] and a 99% fidelity source of partially entangled polarization photonic qubits, we obtain a clear violation of the inequality, excluding a much larger range of measurement dependent local models than would be possible with an adapted Clauser-Horne-Shimony-Holt (CHSH) inequality. It is therefore shown that the measurement independence assumption can be widely relaxed while still demonstrating quantum nonlocality.
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.
Bell tests, as primitive tools to detect nonlocality in bipartite systems, rely on an assumption, i.e., measurement independence. Since ensuring measurement independence in a practical Bell test is difficult, it is crucial to explore the effects of relaxing this assumption. Recently, in the simplest Clauser-Horne-Shimony-Holt Bell (CHSH-Bell) test, Koh et al. [Phys. Rev. Lett. 109, 160404 (2012)] built the relation among measurement dependence, guessing probability, and the maximum value of CHSH-Bell correlation function that an adversary (Eve) can fake. As well, Pope et al. [Phys. Rev. A 88, 032110 (2013)] and Yuan et al. [Phys. Rev. A. 91, 032111 (2015)] settled the same problem in the multiple-run scenario with the general input distribution and the factorizable one, respectively. However, pertinent results in the generalized CHSH-Bell test are still missing. Here, we study this problem and establish the relation among measurement dependence, guessing probability, and the maximum value of the generalized CHSH-Bell correlation function that Eve can fake. Furthermore, we also consider the multiple-run scenario and show the relations in both input distributions. Interestingly, compared with the simplest CHSH-Bell test, we find that it is more difficult for Eve to fake a violation in the generalized CHSH-Bell test in some special cases. We expect that our conclusions will serve as a reference for quantum information processing tasks such as quantum key distribution, randomness expansion, and other tasks in the device-independent framework.
Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters M1 and M2, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.
Quantum nonlocality stands as a resource for device independent quantum information processing (DIQIP). It finds repercussions in applications such as, among others, quantum key distribution and generation of randomness. In this work, we investigate two different approaches to attest nonlocality. First we follow the assumption of limited measurement dependence, i.e., that the measurement settings used in Bell inequality tests or DIQIP are partially influenced by the source of entangled particles and/or by an adversary. Then, we introduce the intermediate assumption of limi- ted detection efficiency, that is, in each run of the experiment, the overall detection efficiency is lower bounded by eta_min > 0. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyse the set of possible correlations that fulfil Measurement Dependence/Limited Detection Locality (MDL/LDL) and show that they necessarily satisfy some linear Bell-like inequalities. In both scenarios, quantum theory predicts the violation of such inequalities for l > 0 in the first case, and eta_min > 0 in the other. We validate these assumptions experimentally via a twin-photon implementation in which two users are provided each with one photon out of a partially entangled pair. On one hand, we show with the first inequality that the measurement independence assumption can be widely relaxed while still demonstrating quantum nonlocality. On the other hand with the second inequality, assuming the switches between the measurement bases are not fully controlled by an adversary, nor by hypothetical local variables, we reveal the nonlocality of the established correlations despite a low overall detection efficiency.
In this dissertation, we explore two alternatives to quantum nonlocality in a single-universe framework: superdeterminism and retrocausality. These models circumvent Bell's theorem by violating the assumption that the hidden variables are uncorrelated with the measurement settings. In Chapter 1, we introduce superdeterminism and prepare the groundwork for our results in Chapter 2. We start from a review of the topic, focussing on the various qualitative criticisms raised against superdeterminism in the literature. We identify the criticism of 'superdeterministic conspiracy' by Bell as the most serious, and introduce nonequilibrium extensions of superdeterministic models in an attempt to formalise the criticism. We study the different properties of these models, and isolate two conspiratorial features. First, the measurement statistics depend on the physical system used to determine the measurement settings. Second, the formal no-signalling constraints are violated although there is no causal connection between the wings. We quantify in Chapter 2 the two conspiratorial features that we found in Chapter 1. We consider a Bell scenario where, in each run and at each wing, the experimenter chooses one of N devices to determine the local measurement setting. We then show that a superdeterministic model of the scenario has to finely tuned such that measurement statistics do not depend on the physical system used to determine the measurement settings. This quantifies the first form of conspiracy identified in Chapter 1 as a fine tuning. We also show that a superdeterministic model of the scenario requires arbitrarily large correlations, quantified in terms of a formal entropy drop and in terms of mutual information, to be set up by the initial conditions. Such correlations are required to ensure that the devices that the hidden variables are correlated with are coincidentally the same as the devices in fact used for every run at each wing. This quantifies the second form of conspiracy identified in Chapter 2 as an arbitrary large correlation. Nonlocal and retrocausal models turn out to be non-conspiratorial according to both approaches, thereby singling out `conspiracy' as a unique, problematic feature of superdeterminism. In Chapter 3, we change tracks to focus on retrocausality. Our results regarding retrocausal models are, in contrast to superdeterminism, positive. We first show how to construct a local, ψ-epistemic hidden-variable model of Bell correlations with wavefunctions in physical space by a retrocausal adaptation of the originally superdeterministic model given by Brans. We show that, in non-equilibrium, the model generally violates no-signalling constraints while remaining local with respect to both ontology and interaction between particles. Lastly, we argue that our model shares some structural similarities with the modal class of interpretations of quantum mechanics. In Chapter 4, focus on the question whether retrocausal models can utilise their relativistic properties to account for relativistic effects on entangled systems. We consider a hypothetical relativistic Bell experiment, where one of the wings experiences time-dilation effects. We show that the retrocausal Brans model, introduced in Chapter 3, can be easily generalised to analyse this experiment, and that it predicts less separation of eigenpackets in the wing experiencing the time-dilation. This causes the particle distribution patterns on the photographic plates to differ between the wings -- an experimentally testable prediction of the model. We discuss the difficulties faced by other hidden variable models in describing this experiment, and their natural resolution in our model due to its relativistic properties. Lastly, we argue that it is not clear at present, due to technical difficulties, if our prediction is reproduced by quantum field theory. We conclude that if it is, then the retrocausal Brans model predicts the same result with great simplicity in comparison. If not, the model can be experimentally tested.