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  • Open Access

Fully Quantum Inflation: Quantum Marginal Problem Constraints in the Service of Causal Inference

Isaac D. Smith1,*, Elie Wolfe2,†, and Robert W. Spekkens2,‡

  • *Contact author: isaac.smith@uibk.ac.at
  • †Contact author: ewolfe@perimeterinstitute.ca
  • ‡Contact author: rspekkens@perimeterinstitute.ca
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PRX Quantum 7, 010351 – Published 13 March, 2026

DOI: https://doi.org/10.1103/244y-nh5n

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 includes the triangle scenario as a special case, for which causal compatibility constraints can be derived.

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Popular Summary

In our modern society, interconnected telecommunication via, e.g., the Internet is ubiquitous. Pursuant to improvements in quantum technologies, there is a current and ongoing global effort to establish a quantum Internet, where communication is mediated by entangled quantum states. These entangled states are typically prepared at certain source nodes of the network before being distributed to the remaining nodes of the network, where further processing may occur. However, if the network has a specific topology, it may not be the case that any desired state can be prepared at the remaining nodes in this way.

In this article, we consider this state preparability question. Specifically, we present several “witnesses” which, like witnesses of quantum entanglement, can be used to demonstrate when a state is unable to be prepared in a certain network. We demonstrate the utility of these witnesses by answering the above question for various states and networks. These witnesses are derived using minimal assumptions: the entangled states distributed by the source nodes can be arbitrary (but finite dimensional). The only other constraints imposed are those of the network topology. Accordingly, due to these minimal assumptions, any resulting statement of state unpreparability is correspondingly strong.

A key contribution of this work is the method for deriving the witnesses mentioned above. There are, however, various questions that remain to be answered. Can stronger witnesses be found? What about a complete set of witnesses characterizing network state preparability exactly?

Article Text

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PRX Quantum 7, 010351– Published 13 March, 2026
  • Received 17 March 2025
  • Accepted 26 November 2025
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