Correlations are ubiquitous in nature and their principled study is of paramount importance in scientific development. The seminal contributions from John Bell offer a framework for analyzing the correlations between the components of quantum mechanical systems and have instigated an experimental tradition which has recently culminated with the Nobel Prize in Physics (2022). In physics, Bell's framework allows the demonstration of the non-classical nature of quantum systems just from the analysis of the observed correlation patterns. Bell's ideas need not be restricted to physics. Our contribution is to show an example of a Bell approach, based on the insight that correlations can be broken down into a part due to common, ostensibly significant causes, and a part due to noise. We employ data from finance (price changes of securities) as an example to demonstrate our approach, highlighting several general applications: first, we demonstrate a new measure of association, informed by the assumed causal relationship between variables. Second, our framework can lead to streamlined Bell-type tests of widely employed models of association, which are in principle applicable to any discipline. In the area of finance, such models of association are Factor Models and the bivariate Gaussian model. Overall, we show that Bell's approach and the models we consider are applicable as general statistical techniques, without any domain specificity. We hope that our work will pave the way for extending our general understanding for how the structure of associations can be analyzed.
Correlations are ubiquitous in nature and their principled study is of paramount importance in scientific development. The seminal contributions from John Bell offer a framework for analyzing the correlations between the components of quantum mechanical systems and have instigated an experimental tradition which has recently culminated with the Nobel Prize in Physics (2022). In physics, Bell's framework allows the demonstration of the non-classical nature of quantum systems just from the analysis of the observed correlation patterns. Bell's ideas need not be restricted to physics. Our contribution is to show an example of a Bell approach, based on the insight that correlations can be broken down into a part due to common, ostensibly significant causes, and a part due to noise. We employ data from finance (price changes of securities) as an example to demonstrate our approach, highlighting several general applications: first, we demonstrate a new measure of association, informed by the assumed causal relationship between variables. Second, our framework can lead to streamlined Bell-type tests of widely employed models of association, which are in principle applicable to any discipline. In the area of finance, such models of association are Factor Models and the bivariate Gaussian model. Overall, we show that Bell's approach and the models we consider are applicable as general statistical techniques, without any domain specificity. We hope that our work will pave the way for extending our general understanding for how the structure of associations can be analyzed.
Explaining observations in terms of cause-and-effect (causal) relations is central to empirical science. However, in his seminal 1964 work [1], John Bell showed that the correlations between entangled quantum systems cannot be explained in terms of, possibly hidden, cause and effect relations. After four decades of experimental effort, Bell's prediction has recently been confirmed in an unambiguous way by a series of loophole-free Bell-test experiments [2-4]. These experiments, however, cannot reveal where exactly quantum mechanics breaks with our classical notion of cause and effect. Specifically, Bell's theorem is based on the central assumptions of realism and local causality. Realism posits that physical systems have real, objective properties, independent of whether we observe them or not, and local causality assumes that causal influences cannot propagate faster than the speed of light.
Quantum mechanics challenges our intuition on the cause-effect relations in nature. Some fundamental concepts, including Reichenbach's common cause principle or the notion of local realism, have to be reconsidered. Traditionally, this is witnessed by the violation of a Bell inequality. But are Bell inequalities the only signature of the incompatibility between quantum correlations and causality theory? Motivated by this question, we introduce a general framework able to estimate causal influences between two variables, without the need of interventions and irrespectively of the classical, quantum, or even postquantum nature of a common cause. In particular, by considering the simplest instrumental scenario-for which violation of Bell inequalities is not possible-we show that every pure bipartite entangled state violates the classical bounds on causal influence, thus, answering in negative to the posed question and opening a new venue to explore the role of causality within quantum theory.
The long-lasting view of entanglement as the characteristic trait of quantum mechanics has been recently challenged by experimental demonstrations of non-quantum entanglement. This motivates a review of the meaning of Bell violations, which have been widely taken to prove the impossibility of a realistic interpretation of quantum mechanics and as a manifestation of its non-local character. This work provides new theoretical evidence for the need of reviewing the meaning of Bell violations, especially when they occur outside the quantum framework. We present a local-realistic model that reproduces quantum predictions concerning Bell tests. We claim that local-realism is fully compatible with correlations that are not of the Bell type and therefore lie outside the scope of Bell's theorem. Most experimental Bell tests involve either spin vectors spanning the Bloch sphere or Stokes vectors spanning the Poincaré sphere. A suitable statistical tool that allows assessing correlations between vectors is given by inner-product-type correlations. Using them, it is possible to reproduce quantum predictions for all Bell states, thereby explaining experimental results of Bell tests within a local-realistic framework.
(English) Understanding the cause and effect relationships behind observed correlations is central to how we reason and interact with the world. Causal relationships help us make sense of the patterns we observe and predict what interventions in nature might lead to a desired outcome. These patterns can be mathematically framed as the joint probability distribution of a set of classical random variables which capture information gathered from the environment. This information may range from abstract data, like survey response statistics, to physical events, such as the probability of triggering a photon detector. A fundamental question is that of causal compatibility: Are the observed correlations compatible with a given causal explanation? A causal explanation can be expressed in terms of causal models, which can be systematically studied with the tools provided by the field of causal inference. Causal models consist of observable random variables with known probability distributions and latent variables with unknown distributions which, together, explain observed correlations through causal influences, that is, functional relationships between the values of these variables. Quantum theory---one of the most accurate theories at a fundamental level---is inherently probabilistic. Measurement results are, therefore, represented as random variables. This naturally leads to causal analysis: Which cause and effect relationships can explain observed measurement statistics in a quantum experiment? One of the simplest quantum experiments is that of two distant parties performing space-like separated, independently chosen measurements on a shared quantum state. In 1964, John Bell showed that in this experiment quantum theory predicts correlations that defy any classical common-cause explanation through a result known as Bell's Theorem. This phenomenon is known as Bell nonlocality. This thesis aims to operationally characterize the fundamental differences between classical and quantum theories within causal scenarios beyond Bell's common-cause scenario. Such an understanding may eventually help integrate quantum phenomena into a coherent, conceptually clear framework of causality. Towards this goal, we explore how classical and quantum causal models diverge in operational tasks in specific causal scenarios. We focus on simple scenarios that go beyond Bell's, while seeking to discover new forms of quantum advantage that are fundamentally different from traditional Bell nonlocality. Our goal is to link these new forms of quantum advantage to different nonclassical features of quantum theory and study their potential applications. A critical component of this research is testing for the causal compatibility of specific correlations with a given causal model. As such, an important part of this thesis is dedicated to expanding and refining the scope of current methods for testing causal compatibility. (Català) Entendre les relacions de causa i efecte darrere de les correlacions observades és central per a la nostra manera de raonar i interactuar amb el món. Les relacions de causa i efecte ens ajuden a donar sentit als patrons que observem i predir quines intervencions en la naturalesa podrien portar a un resultat desitjat. Aquests patrons es poden emmarcar matemàticament com la distribució de probabilitat conjunta d'un conjunt de variables aleatòries clàssiques que capturen informació recollida de l'entorn. Aquesta informació pot variar des de dades abstractes, com estadístiques de respostes d'enquestes, fins a esdeveniments físics, com la probabilitat d'activar un fotodetector. Una pregunta fonamental és la de la compatibilitat causal: Són les correlacions observades compatibles amb una explicació causal donada? Una explicació causal pot expressar-se en termes de models causals, els quals poden ser estudiats sistemàticament amb les eines proporcionades pel camp de la inferència causal. Els models causals consisteixen en variables aleatòries observables amb distribucions de probabilitat conegudes i variables latents amb distribucions desconegudes que, juntes, expliquen les correlacions observades a través d'influències causals, és a dir, relacions funcionals entre els valors d'aquestes variables. La teoria quàntica, una de les teories més precises a nivell fonamental, és inherentment probabilística. Per tant, els resultats de les mesures es representen com a variables aleatòries. Això porta naturalment a l'anàlisi causal: Quines relacions de causa i efecte poden explicar els resultats obtinguts en un experiment quàntic? Un dels experiments quàntics més simples és aquell en què dos laboratoris distants realitzen mesures triades independentment i separades espacialment sobre un estat quàntic compartit. El 1964, John Bell va demostrar que en aquest experiment la teoria quàntica prediu correlacions que desafien qualsevol explicació clàssica en termes de causes comunes. Aquest resultat és conegut com el Teorema de Bell, i el fenomen com no localitat de Bell. Aquesta tesi té com a objectiu caracteritzar operacionalment les diferències fonamentals entre les teories clàssiques i quàntiques dins d'escenaris causals més enllà de l'escenari de causa comuna de Bell. Aquest enteniment eventualment pot ajudar a integrar fenòmens quàntics en un marc de causalitat coherent i conceptualment clar. Per assolir aquest objectiu, explorem com els models causals clàssics i quàntics divergeixen en tasques operacionals en escenaris causals específics. Ens centrem en escenaris simples que van més enllà dels de Bell, mentre busquem descobrir noves formes d'avantatge quàntic que són fonamentalment diferents de la no localitat de Bell tradicional. El nostre objectiu és vincular aquestes noves formes d'avantatge quàntic amb diferents propietats no clàssiques de la teoria quàntica i estudiar les seves aplicacions potencials. Un component crític d'aquesta investigació és provar la compatibilitat causal de correlacions específiques amb un model causal donat. Així doncs, una part important d'aquesta tesi està dedicada a expandir i refinar l'abast dels mètodes actuals per provar la compatibilitat causal. (Español) Entender las relaciones de causa y efecto detrás de las correlaciones observadas es central para nuestra forma de razonar e interactuar con el mundo. Las relaciones de causa y efecto nos ayudan a dar sentido a los patrones que observamos y predecir qué intervenciones en la naturaleza podrían llevar a un resultado deseado. Estos patrones pueden enmarcarse matemáticamente como la distribución de probabilidad conjunta de un conjunto de variables aleatorias clásicas que capturan información recogida del entorno. Esta información puede variar desde datos abstractos, como estadísticas de respuestas de encuestas, hasta eventos físicos, como la probabilidad de activar un fotodetector. Una pregunta fundamental es la de la compatibilidad causal: ¿Son las correlaciones observadas compatibles con una explicación causal dada? Una explicación causal puede expresarse en términos de modelos causales, los cuales pueden ser estudiados sistemáticamente con las herramientas proporcionadas por el campo de la inferencia causal. Los modelos causales consisten en variables aleatorias observables con distribuciones de probabilidad conocidas y variables latentes con distribuciones desconocidas que, juntas, explican las correlaciones observadas a través de influencias causales, es decir, relaciones funcionales entre los valores de estas variables. La teoría cuántica, una de las teorías más precisas a nivel fundamental, es inherentemente probabilística. Por lo tanto, los resultados de las mediciones se representan como variables aleatorias. Esto lleva naturalmente al análisis causal: ¿Qué relaciones de causa y efecto pueden explicar las resultados obtenidos en un experimento cuántico? Uno de los experimentos cuánticos más simples es aquel en el que dos laboratorios distantes realizan mediciones independientemente elegidas y espacialmente separadas sobre un estado cuántico compartido. En 1964, John Bell demostró que en este experimento la teoría cuántica predice correlaciones que desafían cualquier explicación clásica en terminos de causas comunes. Este resultado es conocido como el Teorema de Bell, y el fenómeno como no localidad de Bell. Esta tesis tiene como objetivo caracterizar operacionalmente las diferencias fundamentales entre las teorías clásicas y cuánticas dentro de escenarios causales más allá del escenario de causa común de Bell. Tal entendimiento eventualmente puede ayudar a integrar fenómenos cuánticos en un marco de causalidad coherente y conceptualmente claro. Hacia este objetivo, exploramos cómo los modelos causales clásicos y cuánticos divergen en tareas operacionales en escenarios causales específicos. Nos enfocamos en escenarios simples que van más allá de los de Bell, mientras buscamos descubrir nuevas formas de ventaja cuántica que son fundamentalmente diferentes de la no localidad de Bell tradicional. Nuestro objetivo es vincular estas nuevas formas de ventaja cuántica con diferentes propiedades no clásicas de la teoría cuántica y estudiar sus potenciales aplicaciones. Un componente crítico de esta investigación es probar la compatibilidad causal de correlaciones específicas con un modelo causal dado. Como tal, una parte importante de esta tesis está dedicada a expandir y refinar el alcance de los métodos actuales para probar la compatibilidad causal.
This thesis reports progress in two domains, causal structures and microscopic thermodynamics, both of which are pertinent in the development of quantum technologies. The first part is dedicated to the analysis of causal structure, which encodes the relationship between observed variables, in general restricting the set of possible correlations between them. Our considerations rely on a recent entropy vector method. Based on this, we develop techniques for deriving entropic constraints to differentiate between causal structures. We provide sufficient conditions for entropy vectors to be realisable within a causal structure and derive new, improved necessary conditions in terms of so-called non-Shannon inequalities. We also report that for a family of causal structures, including the bipartite Bell scenario and the bilocal causal structure, entropy vectors are unable to distinguish between classical and quantum causes, in spite of the existence of quantum correlations that are not classically reproducible. Hence, further development is needed in order to understand cause from a quantum perspective. In the second part we explore an axiomatic framework for modelling error-tolerant processes in microscopic thermodynamics. Our axiomatisation allows for the accommodation of finite precision levels, crucial for describing experiments in the microscopic regime. It leads to the emergence of manageable quantities that give insights into the feasibility and expenditure of processes, which for adiabatic processes are shown to be smooth entropy measures. Our framework also leads to thermodynamic behaviour at the macroscopic scale, meaning that for thermodynamic equilibrium states a unique function provides necessary and sufficient conditions for state transformations, like in the traditional second law.
In his seminal paper, Bell [Physics Physique Fizika 1, 195 (1964)] considers the correlations that result from space-like separated measurements on a pair of entangled particles. He uses relativity theory to motivate the Bell causal structure, then shows the existence of quantum correlations that cannot be explained classically within this causal structure. Classical explanations of such quantum correlations are possible in alternative causal structures, for instance, those that allow superluminal causal influences, but, as shown in [New Journal of Physics 17 033002 (2015)], all such alternative explanations require fine tuning (causation without correlation). Here we discuss the existence of spurious quantum correlations --- correlations that look quantum in one causal structure, but have a natural classical explanation in another. More precisely, there are causal structures that admit non-classical quantum correlations, but for which the same correlations have a classical explanation in another causal structure without fine tuning. The realisation in the other causal structure can also be achieved without breaking any natural constraints on the causal structure that follow from relativity theory. However, similarly to non-classical quantum correlations in the Bell causal structure, we find other causal structures with non-classical quantum correlations that do not have a classical causal explanation in any alternative causal structure without fine tuning.