Package of facts and theorems for efficiently generating entanglement criteria for many qubits

TL;DR

This paper introduces a mathematical framework for efficiently constructing multipartite entanglement criteria for many qubits, avoiding complex optimizations by leveraging properties of commutativity graphs, demonstrated with examples including genuine 5-qubit entanglement detection.

Contribution

The paper develops a novel set of theorems that enable the construction of entanglement criteria without optimization over state spaces, simplifying the detection process for many-qubit entanglement.

Findings

Criteria can detect genuine 5-qubit entanglement

Bounds are derived from commutativity graph properties

Method avoids optimization over continuous state sets

Abstract

We present a package of mathematical theorems, which allow to construct multipartite entanglement criteria. Importantly, establishing bounds for certain classes of entanglement does not take an optimization over continuous sets of states. These bonds are found from the properties of commutativity graphs of operators used in the criterion. We present two examples of criteria constructed according to our method. One of them detects genuine 5-qubit entanglement without ever referring to correlations between all five qubits.