Multipartite entanglement of random states of qubits

TL;DR

This paper compares the multipartite entanglement properties of Haar-typical and Hadamard states of n-qubits, highlighting Hadamard states' higher average entanglement and their relevance in finding maximally entangled states.

Contribution

It demonstrates that Hadamard states generally have higher entanglement than Haar-typical states and identifies hypergraph states as promising candidates for maximally entangled states.

Findings

Hadamard states exhibit higher average entanglement than Haar-typical states.

Hypergraph states are structurally simple and more likely to be highly entangled.

Hadamard states are a tractable class for exploring multipartite entanglement.

Abstract

We investigate multipartite entanglement via the statistical properties of pure quantum states of n-qubits. By analyzing the distribution of purity among balanced bipartitions, we compare Haar-typical states, uniformly distributed on the unit sphere of states, with Hadamard states, being characterized by equal weights in the computational basis. We analyze different ensembles of Hadamard states characterized by their phase distributions. Through analytical and numerical calculations, we show that Hadamard states exhibit, on average, a higher degree of entanglement than Haar-typical states. In addition, we show that a particular class of Hadamard states, characterized by real coefficients with alternating signs, known as hypergraph states, appears especially relevant in the search for maximally multipartite entangled states, both for their structural simplicity and the increased likelihood of sampling highly entangled states. These results identify Hadamard states as a tractable yet promising class for exploring multipartite entanglement structures and advancing the characterization of maximally multipartite entangled quantum states.