With the advent of quantum information, the violation of a Bell inequality is used to witness the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's theorem is the existence of experimental "free will," meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signaling model with reduced "free will" and bound the adversary’s capabilities in the task of randomness expansion.
With the advent of quantum information, the violation of a Bell inequality is used to witness the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's theorem is the existence of experimental "free will," meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signaling model with reduced "free will" and bound the adversary’s capabilities in the task of randomness expansion.
Device-independent cryptography provides security in various tasks whilst removing an assumption that cryptographers previously thought of as crucial -- complete trust in the machinations of their experimental apparatus. The theory of Bell inequalities as a proof of indeterminism within nature allows for secure device-independent schemes requiring neither trust in the cryptographers' devices nor reliance on the completeness of quantum mechanics. However, the extreme paranoia attributable to the relaxed assumptions within device independence requires an explicit consideration of the previously assumed ability of the experimenters to freely make random choices. This thesis addresses the so-called `free will loophole', presenting Bell tests and associated cryptographic protocols robust against adversarial manipulation of the random number generators with which measurements in a Bell test are selected. We present several quantitative measures for this experimental free will, otherwise known as measurement dependence. We discuss how an eavesdropper maliciously preprogramming the experimenters' untrusted devices can falsely simulate the violation of a Bell inequality. We also bound the amount of Bell violation achievable within a certain degree of measurement dependence. This analysis extends to device-independent randomness expansion, bounding the guessing probability and estimating the amount of privacy amplification required to distil private randomness. The protocol is secure against either arbitrary no-signalling or quantum adversaries. We also consider device-independent key distribution, studying adversarial models that exploit the free will loophole. Finally, we examine a model correlated between the random number generators and Bell devices across multiple runs of a Bell test. This enables an explicit exposition of the optimal cheating strategy and how the correlations manifest themselves within this strategy. We prove that there remain Bell violations for a sufficiently high, yet non-maximal degree of measurement dependence which cannot be simulated by a classical attack, regardless of how many runs of the experiment those choices are correlated over.
Abstract Self-testing, which refers to device independent characterization of the state and the measurement, has been intensively studied in quantum information theory. One of the implicit assumptions in self-testing theorem is, measurement settings should be chosen randomly and independently by each party. The relaxation of this assumption potentially gives untrusted device providers the power to control the device with no quantum feature, even fake the quantum behavior by using the classical strategies within the device. In this paper, we analyze the effects of the reduced measurements independence on self-testing two-qubit entangled states based on the violation of Bell inequalities. By assuming adversary’s capabilities, we show to achieve the maximum CHSH violation in this case, four different state-measurements preparations should be provided. Without run information one could not really self-test the underlying system in each run anymore, but it must be equal with one of four entangled systems up to local unitaries.
Bell's theorem states that no local realistic explanation of quantum mechanical predictions is possible, in which the experimenter has a freedom to choose between different measurement settings. Within a local realistic picture the violation of Bell's inequalities can only be understood if this freedom is denied. We determine the minimal degree to which the experimenter's freedom has to be abandoned, if one wants to keep such a picture and be in agreement with the experiment. Furthermore, the freedom in choosing experimental arrangements may be considered as a resource, since its lacking can be used by an eavesdropper to harm the security of quantum communication. We analyze the security of quantum key distribution as a function of the (partial) knowledge the eavesdropper has about the future choices of measurement settings which are made by the authorized parties (e.g., on the basis of some quasi-random generator). We show that the equivalence between the violation of Bell's inequality and the efficient extraction of a secure key---which exists for the case of complete freedom (no setting knowledge)---is lost unless one adapts the bound of the inequality according to this lack of freedom.
It is well known that the effect of quantum nonlocality, as witnessed by violation of a Bell inequality, can be observed even when relaxing the assumption of measurement independence, i.e., allowing for the source to be partially correlated with the choices of measurement settings; however, what is the minimal amount of measurement independence needed to observe quantum nonlocality? Here we explore this question and consider models with strong measurement-dependent locality, where measurement choices can be perfectly determined in almost all rounds of the Bell test. Nevertheless, we show that quantum nonlocality can still be observed in this scenario, which we conjecture is minimal within the framework we use. We also discuss potential applications in randomness amplification.
Besides well-known conditions of locality or factorisability, deriving the Bell inequalities requires assuming that the distribution of hidden variables and Alice's and Bob's measurement settings be independent of each other. We show that (analogously to violations of locality due to action at a distance) certain violations of this Measurement Independence assumption can be associated with a notion of signalling in principle, thus making them also testable in principle, and spell out the appropriate conditions. Accordingly, we show that by imposing no-signalling one can prove a version of Bell's theorem that does not require the assumption of Measurement Independence. We discuss the "Schulman model" as an example, as well as lessons for "experimental metaphysics".
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.