# Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions

## TL;DR

This paper introduces the concept of graph-associated entanglement cost for multipartite quantum states, providing conditions for exact and approximate state construction over quantum networks modeled as trees, with detailed asymptotic analysis.

## Contribution

It generalizes entanglement cost to multipartite states and establishes necessary and sufficient conditions for their construction over tree-structured quantum networks.

## Key findings

- Exact construction condition based on Schmidt ranks
- Second-order asymptotic analysis for approximate construction
- Framework applicable to quantum networks modeled as trees

## Abstract

We introduce and analyze graph-associated entanglement cost, a generalization of the entanglement cost of quantum states to multipartite settings. We identify a necessary and sufficient condition for any multipartite entangled state to be constructible when quantum communication between the multiple parties is restricted to a quantum network represented by a tree. The condition for exact state construction is expressed in terms of the Schmidt ranks of the state defined with respect to edges of the tree. We also study approximate state construction and provide a second-order asymptotic analysis.