Fully quantum inflation: quantum marginal problem constraints in the service of causal inference

TL;DR

This paper introduces a fully quantum inflation technique based on the quantum marginal problem to determine if multipartite quantum states are compatible with specific causal structures, advancing quantum causal inference methods.

Contribution

It develops a quantum inflation approach for causal inference, providing a complete classification of three-qubit states relative to the triangle scenario and exploring broader network compatibility.

Findings

Classified pure three-qubit states by compatibility with the triangle scenario

Demonstrated the method's applicability to mixed and higher-dimensional states

Identified conditions where measurement-induced distributions reveal incompatibility

Abstract

Consider the problem of deciding, for a particular multipartite quantum state, whether or not it is realizable in a quantum network with a particular causal structure. This is a fully quantum version of what causal inference researchers refer to as the problem of causal discovery. In this work, we introduce a fully quantum version of the inflation technique for causal inference, which leverages the quantum marginal problem. The primary example by which we illustrate the utility of this method is testing compatibility of tripartite quantum states with the quantum network known as the triangle scenario. We show, in particular, how the method yields a complete classification of pure three-qubit states into those that are and those that are not compatible with the triangle scenario. We also provide some illustrative examples involving mixed states and some where one or more of the systems is higher-dimensional. Furthermore, we examine the question of when the incompatibility of a multipartite quantum state with a causal structure can be inferred from the incompatibility of a joint probability distribution induced by implementing measurements on each subsystem. Finally, we present a family of networks, which include the triangle scenario as a special case, for which causal compatibility constraints can be derived.