Causal Inference & Quantum Foundations Workshop

Quantum causality is an emerging field of study which has the potential to greatly advance our understanding of quantum systems. In this paper, we put forth a theoretical framework for merging quantum information science and causal inference by exploiting entropic principles. For this purpose, we leverage the tradeoff between the entropy of hidden cause and the conditional mutual information of observed variables to develop a scalable algorithmic approach for inferring causality in the presence of latent confounders (common causes) in quantum systems. As an application, we consider a system of three entangled qubits and transmit the second and third qubits over separate noisy quantum channels. In this model, we validate that the first qubit is a latent confounder and the common cause of the second and third qubits. In contrast, when two entangled qubits are prepared and one of them is sent over a noisy channel, there is no common confounder. We also demonstrate that the proposed approach outperforms the results of classical causal inference for the Tubingen database when the variables are classical by exploiting quantum dependence between variables through density matrices rather than joint probability distributions.

According to recent new definitions, a multi-party behavior is genuinely multipartite nonlocal (GMNL) if it cannot be modeled by measurements on an underlying network of bipartite-only nonlocal resources, possibly supplemented with local (classical) resources shared by all parties. Three experimental results published in 2022 provide initial evidence, subject to postselection-related assumptions, for the existence of behaviors meeting these definitions of GMNL. The new definitions of GMNL differ on whether to allow entangled measurements upon, and/or superquantum behaviors among, the underlying bipartite resources when classifying behaviors as​only bipartite nonlocal. I will discuss the interrelationships of these choices in three-party quantum networks, and present a behavior in the simplest nontrivial multi-partite measurement scenario (3 parties, 2 measurement settings, and 2 outcomes) that (A) cannot be simulated in a bipartite network prohibiting both entangled measurements and superquantum resources, (B) can be simulated with bipartite-only quantum states allowing for an entangled quantum measurement (indicating an approach to device independent certification of entangled measurements with fewer settings than in previous protocols), and surprisingly (C) can be simulated with bipartite-only superquantum states (Popescu-Rohrlich boxes) while maintaining a prohibition on entangled measurements. It turns out that other behaviors previously studied as device-independent witnesses of entangled measurements can also be simulated in the manner of (C), posing a challenge to a theory-independent understanding of entangled measurements as an observable phenomenon distinct from bipartite nonlocality.

"Many relationships in causality, statistics or probability theory can be expressed as conditional independence relations between the occurring random variables. Since the invention of the notion of conditional independence one aim was to be able to also express such relationship between random and non-random variables, like the parameters of a stochastic model, the input variables of a probabilistic program or intervention variables in a causal model. Over time several different versions of such extended conditional independence notion have been proposed, each coming with their own advantages and disadvantages, oftentimes limited to certain subclasses of random variables like discrete variables or ones with densities. In this talk we present another such notion of conditional independence, which can easily be expressed in measure-theoretic generality and even in categorical probability. We will study its expressivity, present its (convenient) properties, and relate it to other notions of conditional independence."

Explaining the natural world through cause-and-effect relations is the fundamental principle of science. Although a classical theory of causality has been recently introduced, enabling us to model causation across diverse research fields, it is crucial to examine which aspects of it require modification or abandonment to also comprehend causality in the quantum world. To address this question, we will investigate paradigmatic scenarios, including the double slit, Bell's theorem and generalizations to quantum networks, also exploring recent experimental advancements.

We investigate the problem of bounding counterfactual queries from an arbitrary collection of observational and experimental distributions and qualitative knowledge about the underlying data-generating model represented in the form of a causal diagram. We show that all counterfactual distributions in an arbitrary structural causal model (SCM) with finite discrete endogenous variables could be generated by a family of SCMs with the same causal diagram where unobserved (exogenous) variables are discrete with a finite domain. Utilizing this family of SCMs, we translate the problem of bounding counterfactuals into that of polynomial programming whose solution provides optimal bounds for the counterfactual query.

The class of problems in causal inference which seeks to isolate causal correlations solely from observational data even without interventions has come to the forefront of machine learning, neuroscience and social sciences. As new large scale quantum systems go online, it opens interesting questions of whether a quantum framework exists on isolating causal correlations without any interventions on a quantum system. We put forth a theoretical framework for merging quantum information science and causal inference by exploiting entropic principles. At the root of our approach is the proposition that the true causal direction minimizes the entropy of exogenous variables in a non-local hidden variable theory. The proposed framework uses a quantum causal structural equation model to build the connection between two fields: entropic causal inference and the quantum marginal problem. First, inspired by the definition of geometric quantum discord, we fill the gap between classical and quantum conditional density matrices to define quantum causal models. Subsequently, using a greedy approach, we develop a scalable algorithm for quantum entropic causal inference unifying classical and quantum causality in a principled way. We apply our proposed algorithm to an experimentally relevant scenario of identifying the subsystem impacted by noise starting from an entangled state. This successful inference on a synthetic quantum dataset can have practical applications in identifying originators of malicious activity on future multi-node quantum networks as well as quantum error correction. As quantum datasets and systems grow in complexity, our framework can play a foundational role in bringing observational causal inference from the classical to the quantum domain.

For binary instrumental variable models, there seems to be a long-standing gap between two sets of bounds on the average treatment effect: the stronger Balke–Pearl ("sharp") bounds versus the weaker Robins–Manski ("natural") bounds. In the literature, the Balke–Pearl bounds are typically derived under stronger assumptions, i.e., either individual exclusion or joint exogeneity, which are untestable cross-world statements, while the natural bounds only require testable assumptions. In this talk, I show that the stronger bounds are justified by the existence of a latent confounder. In fact, the Balke–Pearl bounds are sharp under latent confounding and stochastic exclusion. The "secret sauce" that closes this gap is a set of CHSH-type inequalities that generalize Bell's (1964) inequality.

"Is there a complete semi-definite programming hierarchy for quantum causal problems? We divide the question into two parts. First: Can quantum causal problems be expressed as polynomial optimization problems (this talk). Second: Can this class of polynomial optimizations be solved by means of SDPs (Laurens' talk). The optimizations we consider here are ""polynomial"" in two ways. They are over the unknown observable algebra of the hidden systems, which are specified by non-commutative polynomials in a set of generators. But they also involve independence constraints, which are commutative polynomials in the state. A hierarchy of such polynomial tests is complete if one can construct a quantum model for any observed distribution that passes all of them. We've recently had some success in finding such constructions, but also ran into problems in the general case [1, 2]. I give a high-level presentation of the state of the play. [1] https://arxiv.org/abs/2110.14659 [2] https://arxiv.org/abs/2212.11299"

"Many relevant tasks in Quantum Information processing can be expressed as polynomial optimization problems over states and operators. In the earlier talk by David, we saw that this is also the case for certain (quantum) causal compatibility and causal optimization problems. This talk will focus on several closely related semidefinite programming (SDP) hierarchies that have recently been shown to be complete for such polynomial optimization problems [arxiv:2110.14659, 2212.11299, 2301.12513]. We give a high-level overview of the techniques and mathematics that are needed for proving such statements. In particular, we will see a version of a Quantum De Finetti theorem, as well as a sketch of a constructive proof of convergence for the SDP hierarchies. Afterwards, these results are linked back to the causal compatibility problem to conclude that such SDP hierarchies are complete for a certain type of causal structures known as tree networks."

In a recent paper, Chaturvedi et al considered the interesting idea of routed Bell experiments. These are Bell experiments where Bob can measure his quantum particles at two distinct locations, one close to the source and another far away. This can be accomplished in the lab by using a switch that directs Bob's quantum particle either to the nearby measurement device or to the distant one, depending on a classical input chosen by Bob. Chaturvedi et al argue that there exists in such experiments a tradeoff between short-range and long-range correlations and that high-quality CHSH tests close to the source (which are achievable with current technology) lower the requirements for witnessing nonlocality faraway from the source, and in particular increase their tolerance to particle losses. We critically review their results and present a simple counterexample to it. We then introduce a class of hybrid quantum-classical models, which we refer to as "short-range quantum models". These models suitably capture the tradeoff between short-range and long-range correlations in routed Bell experiments. Using our definition, we explore new nonlocal tests in which high-quality short-range correlations lead to weakened conditions for long-range tests. Although we do find improvements, they are significantly smaller than those claimed by CVP.

There has been recent interest in extending the concept of contextuality to cases of disturbance or inconsistent connectedness. This talk will describe an approach using probabilistic causal models, which generalize the hidden-variables models of Bell and Kochen & Specker, following recent work by Cavalcanti. I first prove an equivalence between three conditions on an arbitrary measurement system: (1) existence of a model minimizing all causal influences of context upon measurement outcomes, (2) prohibition of a form of "hidden" causal influence, and (3) noncontextuality as defined in the Contextuality-by-Default (CbD) theory of Dzhafarov and Kujala. The no-hidden-influence principle thus confers a physical interpretation to CbD-contextuality, paralleling Bell's local causality and Kochen & Specker's classical embeddability. I then extend this analysis to other causal graph topologies, showing that different graphs yield different notions of contextuality, but only the one corresponding to CbD agrees with traditional contextuality when restricted to non-disturbing systems.

"Linear structural equation models relate random variables of interest via a linear equation system that features stochastic noise. The models are naturally represented by directed graphs whose edges indicate non-zero coefficients in the linear equations. In this talk I will report on progress on combinatorial conditions for parameter identifiability in models with latent (i.e., unobserved) variables. Identifiability holds if the coefficients associated with the edges of the graph can be uniquely recovered from the covariance matrix they define. Paper: https://doi.org/10.1214/22-AOS2221 or https://arxiv.org/abs/2201.04457"