Quantum Foundations

Based on the immense popularity of causal Bayesian networks and structural causal models, one might expect that these representations are appropriate to describe the causal semantics of any real-world system, at least in principle. In this talk, I will argue that this is not the case, and motivate the study of more general causal modeling frameworks. In particular, I will discuss bipartite graphical causal models. Real-world complex systems are often modelled by systems of equations with endogenous and independent exogenous random variables. Such models have a long tradition in physics and engineering. The structure of such systems of equations can be encoded by a bipartite graph, with variable and equation nodes that are adjacent if a variable appears in an equation. I will show how one can use Simon’s causal ordering algorithm and the Dulmage-Mendelsohn decomposition to derive a Markov property that states the conditional independence for (distributions of) solutions of the equations in terms of the bipartite graph. I will then show how this Markov property gives rise to a do-calculus for bipartite graphical causal models, providing these with a refined causal interpretation.

In the Statistics literature there are three main frameworks for causal modeling: counterfactuals (aka potential outcomes), non-parametric structural equation models (NPSEMs) and graphs (aka path diagrams or causal Bayes nets). These approaches are similar and, in certain specific respects, equivalent. However, there are important conceptual differences and each formulation has its own strengths and weaknesses. These divergences are of relevance both in theory and when the approaches are applied in practice. This talk will introduce the different frameworks, and describe, through examples, both the commonalities and dissimilarities. In particular, we will see that the “default” assumptions within these frameworks lead to different identification results when quantifying mediation and, more generally, path-specific effects.

Can the effectiveness of a medical treatment be determined without the expense of a randomized controlled trial? Can the impact of a new policy be disentangled from other factors that happen to vary at the same time? Questions such as these are the purview of the field of causal inference, a general-purpose science of cause and effect, applicable in domains ranging from epidemiology to economics. Researchers in this field seek in particular to find techniques for extracting causal conclusions from statistical data. Meanwhile, one of the most significant results in the foundations of quantum theory—Bell’s theorem—can also be understood as an attempt to disentangle correlation and causation. Recently, it has been recognized that Bell’s result is an early foray into the field of causal inference and that the insights derived from almost 60 years of research on his theorem can supplement and improve upon state-of-the-art causal inference techniques. In the other direction, the conceptual framework developed by causal inference researchers provides a fruitful new perspective on what could possibly count as a satisfactory causal explanation of the quantum correlations observed in Bell experiments. Efforts to elaborate upon these connections have led to an exciting flow of techniques and insights across the disciplinary divide. This tutorial will highlight some of what is happening at the intersection of these two fields.

The standard perspective on subsystems in quantum theory is a bottom-up, compositional one: one starts with individual "small" systems, viewed as primary, and composes them together to form larger systems. The top-down, decompositional perspective goes the other way, starting with a "large" system and asking what it means to partition it into smaller parts. In this talk, I will 1/ argue that the adoption of the top-down perspective is the key to progress in several current areas of foundational research; and 2/ present an integrated mathematical framework for partitions into three or more subsystems, using sub-C* algebras. Concerning the first item, I will explain how the top-down perspective becomes crucial whenever the way in which a quantum system is partitioned into smaller subsystems is not unique, but might depend on the physical situation at hand. I will display how that precise feature lies at the heart of a flurry of current hot foundational topics, such as quantum causal models, Wigner's friend scenarios, superselection rules, quantum reference frames, and debates over the implementability of the quantum switch. Concerning the second item, I will argue that partitions in (finite-dimensional) quantum theory can be naturally pinned down using sub-C* algebras. Building on simple illustrative examples, I will discuss the often-overlooked existence of non-factor C*-algebras, and how it leads to numerous subtleties -- in particular a generic failure of local tomography. I will introduce a sound framework for quantum partitions that overcomes these challenges; it is the first top-down framework that allows to consider three or more subsystems. Finally, as a display of this framework's technical power, I will briefly present how its application to quantum causal modelling unlocked the proof that all 1D quantum cellular automata admit causal decompositions.

(This is joint work with Octave Mestoudjian and Pablo Arrighi. This talk is complementary to my Causalworlds 2024 presentation, which will focus on the issue of causal decompositions.)