A convergent inflation hierarchy for quantum causal structures

TL;DR

This paper introduces a convergent quantum inflation hierarchy for causal structures, providing a provably complete method to determine compatibility of quantum distributions with causal models using advanced de Finetti Theorems.

Contribution

It develops the first provably convergent quantum inflation hierarchy, extending classical techniques and generalizing de Finetti Theorems to broader quantum settings.

Findings

Hierarchy is proven to be convergent.

Introduces a Quantum de Finetti Theorem for general tensor products.

Extends classical inflation techniques to quantum causal structures.

Abstract

A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states - however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. From a technical point of view, convergence proofs are built on de Finetti Theorems, which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of $C^*$-algebras, generalizing previous work that was restricted to minimal tensor products.