Unifying Framework for Relaxations of the Causal Assumptions in Bell’s Theorem

Abstract

Bell's Theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions -e.g. "locality" or "measurement independence" -have to be relaxed in order to allow for a classical description of such experiments. Here, we develop a conceptual and computational framework for treating this problem. We employ the language of Bayesian networks to systematically construct alternative causal structures and bound the degree of relaxation using quantitative measures that originate from the mathematical theory of causality. The main technical insight is that the resulting problems can often be expressed as computationally tractable linear programs. We demonstrate the versatility of the framework by applying it to a variety of scenarios, ranging from relaxations of the measurement independence, locality and bilocality assumptions, to a novel causal interpretation of CHSH inequality violations.The paradigmatic Bell experiment [1] involves two distant observers, each with the capability to perform one of two possible experiments on their shares of a joint system. Bell observed that even absent of any detailed information about the physical processes involved, the causal structure of the setup alone implies strong constraints on the correlations that can arise from any classical description [2]. The physically wellmotivated causal assumptions are: (i) measurement independence: experimenters can choose which property of a system to measure, independently of how the system has been prepared; (ii) locality: the results obtained by one observer cannot be influenced by any action of the other (ideally space-like separated) experimenter. The resulting constraints are Bell's inequalities [1]. Quantum mechanical processes subject to the same causal structure can violate these constraints -a prediction that has been abundantly verified experimentally [3][4][5][6][7]. This effect is commonly referred to as quantum nonlocality.It is now natural to ask how stable the effect of quantum non-locality is with respect to relaxations of the causal assumptions. Which "degree of measurement dependence", e.g., is required to reconcile empirically observed correlations with a classical and local model? Such questions are not only, we feel, of great relevance to foundational questions -they are also of interest to practical applications of non-locality, e.g. in cryptographic protocols. Indeed, eavesdroppers can (and do [8]) exploit the failure of a given cryptographic device to be constrained by the presumed causal structure to compromise its security. At the same time, it will often be difficult to ascertain that causal assumptions hold exactly -which makes it important to develop a systematic quantitative theory.Several variants of this question have recently attracted considerable attention [9][10][11][12][13][14][15][16][17][18][19][20]. For example, measurement dependence has been found ...

Experimental violation of local causality in a quantum network

“…This allows us to violate the bilocal causality inequality proposed in ref. 19 and further developed in refs 20, 22, 23, 24, 25:…”

“…1c), which introduce additional structure to the set of classically allowed correlations. In fact, there are quantum correlations that can emerge in networks that, while admitting a LHV description, are incompatible with any classical description where the independence of the sources is considered19202122232425. For instance, a network with two independent sources allow for the emergence of a different kind of non-local correlations violating the so-called bilocal causality assumption1920.…”

Causality in the Quantum World

“…Several researchers recognized that the formalism of classical causal models could provide a unifying framework in which to pose the problem of deriving Bell-type constraints, and that this framework might be extended to address the problem of deriving constraints on the correlations that can be obtained with quantum resources [2,10,11,62]. Note that such constraints are expressed entirely in terms of classical settings and classical outcomes of measurements.…”

“…Although our main motivation for developing quantum causal models is the possibility of finding a satisfactory (i.e., non-fine-tuned) causal explanation of Bell inequality violations [2,10], they are also likely to have practical applications. For instance, finding quantum-classical separations in the correlations achievable in novel causal scenarios might lead to new device-independent protocols [11], such as randomness extraction and secure key distribution.…”

Explaining quantum correlations through evolution of causal models

“…Building off the work of Chaves et al [29], we make all this concrete through a quantification of the relaxation of each assumption in the context of causal models. The task of minimizing the amount of the relaxation is a multi-objective optimization problem.…”