# Almost all four-particle pure states are determined by their two-body marginals

## TL;DR

This paper proves that most pure states of four particles are uniquely identified by their two-body marginals, with some symmetric states sharing marginals and remaining undetermined, extending to larger systems.

## Contribution

It demonstrates that generic four-particle pure states are determined by a small set of two-body marginals, and generalizes the result to larger systems with specific exceptions.

## Key findings

- Most four-particle pure states are uniquely determined by their two-body marginals.
- Certain symmetric states share marginals and are not uniquely determined.
- The result extends to larger systems with specific undetermined states.

## Abstract

We show that generic pure states (states drawn according to the Haar measure) of four particles of equal internal dimension are uniquely determined among all other pure states by their two-body marginals. In fact, certain subsets of three of the two-body marginals suffice for the characterization. We also discuss generalizations of the statement to pure states of more particles, showing that these are almost always determined among pure states by three of their $(n-2)$-body marginals. Finally, we present special families of symmetric pure four-particle states that share the same two-body marginals and are therefore undetermined. These are four-qubit Dicke states in superposition with generalized GHZ states.