# Feynman graphs and the large dimensional limit of multipartite entanglement

## TL;DR

This paper investigates the properties of multipartite entanglement in high-dimensional quantum systems, using Feynman graphs and statistical mechanics to analyze purity distribution and its limits as system dimension grows.

## Contribution

It introduces a novel approach linking multipartite entanglement optimization to high-temperature expansions and Feynman graph techniques, especially in the large dimension limit.

## Key findings

- Series expansion of purity distribution converges.
- Behavior of series terms analyzed as dimension increases.
- Lower bounds of purity can be approached for all system sizes.

## Abstract

We are interested in the properties of multipartite entanglement of a system composed by $n$ $d$-level parties (qudits). Focussing our attention on pure states we want to tackle the problem of the maximization of the entanglement for such systems. In particular we effort the problem trying to minimize the purity of the system. It has been shown that not for all systems this function can reach its lower bound, however it can be proved that for all values of $n$ a $d$ can always be found such that the lower bound can be reached. In this paper we examine the high-temperature expansion of the distribution function of the bipartite purity over all balanced bipartition considering its optimization problem as a problem of statistical mechanics. In particular we prove that the series characterizing the expansion converges and we analyze the behavior of each term of the series as $d\to \infty$.