Causal Inference & Quantum Foundations Workshop

Apr 17, 2023, 8:30 a.m. → Apr 21, 2023, 5:00 p.m. America/Toronto

PI/4-405 - Bob Room (Perimeter Institute for Theoretical Physics)

Elie Wolfe (Perimeter Institute), Ilya Shpitser (Johns Hopkins University), Robert Spekkens (Perimeter Institute)

Recently we have seen exciting results at the intersection of quantum foundations and the statistical analysis of causal hypotheses by virtue of the centrality of latent variable models to both fields.

In this workshop we will explore how academics from both sides can move the shared frontiers forward. Towards that end, we are including extensive breakout collaboration opportunities in addition to formal presentations. In order to make concrete progress on problems pertinent to both communities, we have selected the topic of causal models with restricted cardinality of the latent variables as a special focus for this workshop. 

Sponsorship for this workshop has been provided by:

https://pirsa.org/C23017
 

Territorial Land Acknowledgement

Perimeter Institute acknowledges that it is situated on the traditional territory of the Anishinaabe, Haudenosaunee, and Neutral peoples.

Perimeter Institute is located on the Haldimand Tract. After the American Revolution, the tract was granted by the British to the Six Nations of the Grand River and the Mississaugas of the Credit First Nation as compensation for their role in the war and for the loss of their traditional lands in upstate New York. Of the 950,000 acres granted to the Haudenosaunee, less than 5 percent remains Six Nations land. Only 6,100 acres remain Mississaugas of the Credit land. 

We thank the Anishinaabe, Haudenosaunee, and Neutral peoples for hosting us on their land.

Monday, April 17

9:00 a.m. Registration

1 Welcome and Opening Remarks

Speaker: Elie Wolfe (Perimeter Institute)

2 Tutorial 1

Speaker: Robert Spekkens (Perimeter Institute)

11:00 a.m. Coffee Break

3 Graphical models: fundamentals, origins, and beyond

The lecture will give a brief introduction to graphical models, their origins in Physics, Genetics, and Econometrics, their modern usages, and some future perspectives.

Speaker: Steffen Lauritzen (University of Copenhagen)

4 Towards standard imsets for maximal ancestral graphs

"Imsets, introduced by Studený (see Studený, 2005 for details), are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice when applied to directed acyclic graph (DAG) models. In particular, the standard imset for a DAG is in one-to-one correspondence with the independence model it induces, and hence is a label for its Markov equivalence class. We present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, which have directed and bidirected edges, using the parameterizing set representation of Hu and Evans (2020). By construction, our imset also represents the Markov equivalence class of the MAG.

We show that for many such graphs our proposed imset defines the model, though there is a subclass of graphs for which the representation does not. We prove that it does work for MAGs that include models with no adjacent bidirected edges, as well as for a large class of purely bidirected models. If there is time, we will also discuss applications of imsets to structure learning in MAGs.

This is joint work with Zhongyi Hu (Oxford).

References

Z. Hu and R.J. Evans, Faster algorithms for Markov equivalence, In Proceedings for the 36th Conference on Uncertainty in Artificial Intelligence (UAI-2020), 2020.

M. Studený, Probabilistic Conditional Independence Structures, Springer-Verlag, 2005."

Speaker: Robin Evans (University of Oxford)

12:30 p.m. Lunch

5 Tutorial 2

Speaker: Ilya Shpitser (Johns Hopkins University)

3:00 p.m. Coffee Break

6 Correlations from joint measurements in boxworld and applications to information processing

Quantum measurements have been a central topic of research in quantum theory for many years. In the context of causal structures and communication over networks, we are often particularly interested in local measurements of subsystems of a multi-partite system and classical processing of their inputs and outcomes. Formally, this processing can often be described by means of maps that are known as wirings. These wirings are furthermore interesting for the analysis of generalized probabilistic theories, as they are shared by all of them. In this work, we explicitly characterise all possible mulitpartite measurements in the generalised probabilistic theory box-world for various numbers of parties n with systems characterised by n_i fiducial measurements (which can be thought of as inputs here) and n_o outcomes, for small n, n_i, n_o. This includes all n-party n_i-input, n_o-outcome wirings. For n > 2, we further classify these measurements into three classes: wirings, deterministic non-wiring type and non-deterministic non-wiring type measurements. We explore advantages of these different types of measurements over previous protocols in the context of non-locality distillation and state-distingishability. We further find examples of non-locality without entanglement (contrary to previous claims) and a relation of these measurements to classical process matrices.

Speaker: Mirjam Weilenmann (Institute for Quantum Optics and Quantum Information)

7 Observational Equivalences Between Causal Structures with Latent Variables

"If one only performs experiments involving passive observations, in general there are multiple causal structures that can explain the same set of distributions over the observed variables. In this case, we say that these causal structures are observationally equivalent. In this work, we explore all the known techniques for proving observational equivalence or inequivalence, as well as some original ones.
Even if the existing rules are not enough to achieve the full classification of the causal structures with four observed variables, our results get close to such classification and show that admitting inequality constraints is a generic feature among structures with four observed variables."

Speaker: Marina Maciel Ansanelli (Perimeter Institute)

8 Entropic Inequality Constraints from e-separation Relations in Directed Acyclic Graphs with Hidden Variables

When some variables in a directed acyclic graph (DAG) are hidden, a notoriously complicated set of constraints on the distribution of observed variables is implied. In this talk, we present inequality constraints implied by graphical criteria in hidden variable DAGs. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy. For DAGs that exhibit e-separation relations, we present entropic inequality constraints and we show how they can be used to learn about the true causal model from an observed data distribution (arXiv:2107.07087).

Speaker: Beata Zjawin (Gdansk University)

Tuesday, April 18

9 Communication Genuine Multipartite Nonlocality as a benchmark for large nonclassicality

"Quantum computing requires the ability to manipulate large nonclassical quantum systems. As we are far from any useful quantum computing advantage, certifying this ability is an important benchmark to assess progress toward this goal. This can be done using the nonlocal nature of quantum correlations, which allows to certify a non-trusted experimental apparatus from its input/output behaviour in a device independent way. It first requires to introduce the concept of Genuine Multipartite Nonlocality (GMNL) of size n, which designate systems which nonlocality cannot be understood an obtained from many states composed of n − 1 (or less) constituents.

The first historical definition of GMNL, proposed by Svetlichny, is ill-defined when used to assess the large nonclassical nature of quantum systems, as it predicts that maximal GMNL states can be obtain from bipartite sources only. A more appropriate re-definition of that concept, called LOSR-GMNL, was proposed recently [arXiv:2105.09381]. However, it is not satisfactory in all experimental situations, as it cannot (by design) capture potential communications between the systems which could occur in some realistic experimental systems (e.g., many-body systems) – which Svetlichny definition captures in a naïve way.

In this talk, I will propose a new alternative re-definition solving this issue, called Communication-Genuine Multipartite Nonlocality of length t (C-GMNL). It is based on a model inspired from synchronous distributed computing, that involves t communications steps along a graph.

I will show that (i) the GHZ state is maximally nonlocal according to this C-GMNL definition, (ii) the cluster state is trivial in this C-GMNL definition but that (iii) the cluster state is maximally difficult in the LOSR-GMNL definition. Hence, some complicated LOSR-GMNL states become trivial when a small amount of communication is allowed.

Based on a joint work in preparation with Xavier Coiteux-Roy, Owidiusz Makuta, Fionnuala Curran, Remigiusz Augusiak."

Speaker: Marc-Olivier Renou (Institute of Photonic Sciences)

10 Separation of quantum, spatial quantum and approximate quantum correlations

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model and the dimension of the underlying Hilbert spaces, various notions of sets of quantum correlations can be defined. This talk is devoted to the separations of some of these sets via simple ideas in quantum information theory, namely self-testing and entanglement embezzlement.

Speaker: Salman Beigi (Institute for Research in Fundamental Sciences)

10:30 a.m. Coffee Break

11 Causal Scenarios: the Interesting, the Boring and the Elusive

I will sketch the current state of play with classifying causal scenarios (aka DAGs with latent variables). Some are interesting: the classical correlations are constrained by non-trivial inequalities such as Bell’s. Some are boring: the classical correlations are constrained only by observable conditional independencies. Some we still don’t know. Along the way I will mention joint work with Joe Henson, Ray Lal, Shashaank Khanna, Marina Ansanelli and Elie Wolfe, and disjoint work by Robin Evans.

Speaker: Matthew Pusey (University of York)

11:30 a.m. Breakout Groups

12:30 p.m. Lunch

12 Causal Discovery via Common Entropy

Distinguishing causation from correlation from observational data requires assumptions. We consider the setting where the unobserved confounder between two observed variables is simple in an information-theoretic sense, captured by its entropy. When the observed dependence is not due to causation, there exists a small-entropy variable that can make the observed variables conditionally independent. The smallest such entropy is known as common entropy in information theory. We extend this notion to Renyi common entropy by minimizing the Renyi entropy of the latent variable. We establish identifiability results with Renyi-0 common entropy, and a special case of (binary) Renyi-1 common entropy. To efficiently compute common entropy, we propose an iterative algorithm that can be used to discover the trade-off between the entropy of the latent variable and the conditional mutual information of the observed variables. We show that our algorithm can be used to distinguish causation from correlation in such simple two-variable systems. Additionally, we show that common entropy can be used to improve constraint-based methods such as the PC algorithm in the small-sample regime, where such methods are known to struggle. We propose modifying these constraint-based methods to assess if a separating set found by these algorithms is valid using common entropy. We finally evaluate our algorithms on synthetic and real data to establish their performance.

Speaker: Murat Kocaoglu (Purdue University)

13 Quantum causal inference in the presence of hidden common causes: An entropic approach

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.

Speaker: Mohammad Ali Javidian (Appalachian State University)

3:00 p.m. Coffee Break

3:30 p.m. Breakout Groups

Wednesday, April 19

14 A Hierarchy of Multi-Party Nonlocal Effects

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.

Speaker: Peter Bierhorst (University of New Orleans)

15 Conditional Independence - Revisited

"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."

Speaker: Patrick Forre (Universiteit van Amsterdam)

10:30 a.m. Coffee Break

11:00 a.m. Breakout Groups

12:30 p.m. Lunch

16 A quantum tale of causes and effects

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.

Speaker: Rafael Chaves (Federal University of Rio Grande do Norte)

3:00 p.m. Coffee Break

17 Bounding counterfactual distributions in discrete structural causal models

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.

Speaker: Jin Tian (Iowa State University)

18 Quantum entropic causal inference

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.

Speakers: Vaneet Aggarwal (Purdue University), Zubin Jacob (Purdue University)

19 Latent variable justifies the stronger instrumental variable bounds

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.

Speaker: Richard Guo (University of Cambridge)

5:00 p.m. Banquet

Thursday, April 20

20 Is causal optimization polynomial optimization?

"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"

Speaker: David Gross (University of Cologne)

21 SDP approaches for quantum polynomial optimization

"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."

Speaker: Laurens Ligthart (University of Cologne)

10:30 a.m. Coffee Break

11:00 a.m. Breakout Groups

12:30 p.m. Lunch

22 Certifying long-range quantum correlations through routed Bell experiments

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.

Speaker: Stefano Pironio (Université Libre de Bruxelles)

23 Causal-model approach to extended contextuality

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.

Speaker: Matt Jones (University of Colorado)

3:00 p.m. Coffee Break

3:30 p.m. Breakout Groups

Friday, April 21

24 Half-Trek Criterion for Identifiability of Latent Variable Models

"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"

Speaker: Mathias Drton (Technical University Munich)

25 Some applications of Causal Inference in the real world

Causal reasoning is vital for effective reasoning in many domains, from healthcare to economics. In medical diagnosis, for example, a doctor aims to explain a patient’s symptoms by determining the diseases causing them. This is because causal relations, unlike correlations, allow one to reason about the consequences of possible treatments and to answer counterfactual queries. In this talk I will present two recent causal inference projects done with my collaborators deriving new algorithms to solve problems that arise when applying causal inference in the real world.

Speaker: Ciaran Gilligan-Lee (Spotify and University College London)

10:30 a.m. Coffee Break

11:00 a.m. Breakout Groups

12:30 p.m. Lunch

26 Panel Discussion

Speaker: Sonia Markes (University of Toronto)

3:00 p.m. Coffee Break

3:30 p.m. Breakout Groups