Entanglement transitivity problems

TL;DR

This paper investigates how marginal quantum states can reveal new entanglement information, demonstrating transitivity properties and providing conditions for such phenomena in multi-qubit systems.

Contribution

It introduces the concept of entanglement transitivity, proves its existence in large systems, and characterizes conditions for transitivity in tripartite states with specific marginals.

Findings

Transitivity of entanglement can occur in large multi-qubit systems.

Unique global states can be reconstructed from certain two-qubit marginals.

Entanglement transitivity is common among marginals from pure states in tripartite systems.

Abstract

One of the goals of science is to understand the relation between a whole and its parts, as exemplified by the problem of certifying the entanglement of a system from the knowledge of its reduced states. Here, we focus on a different but related question: can a collection of marginal information reveal new marginal information? We answer this affirmatively and show that (non-) entangled marginal states may exhibit (meta)transitivity of entanglement, i.e., implying that a different target marginal must be entangled. By showing that the global $n$-qubit state compatible with certain two-qubit marginals in a tree form is unique, we prove that transitivity exists for a system involving an arbitrarily large number of qubits. We also completely characterize -- in the sense of providing both the necessary and sufficient conditions -- when (meta)transitivity can occur in a tripartite scenario when the two-qudit marginals given are either the Werner states or the isotropic states. Our numerical results suggest that in the tripartite scenario, entanglement transitivity is generic among the marginals derived from pure states.