A convergent inflation hierarchy for 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 $C^{\ast }$-algebras, generalizing previous work that was restricted to minimal tensor products.

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References in corpus (11)

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Cited by in corpus (10)

• Semidefinite programming relaxations for quantum correlations
• Partial self-testing and randomness certification in the triangle network
• Bounding the joint numerical range of Pauli strings by graph parameters
• The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario
• State polynomials: positivity, optimization and nonlinear Bell inequalities
• Topologically Robust Quantum Network Nonlocality
• Two convergent NPA-like hierarchies for the quantum bilocal scenario
• Symmetric observations without symmetric causal explanations
• Positivity of state, trace, and moment polynomials, and applications in quantum information
• Fully quantum inflation: quantum marginal problem constraints in the service of causal inference