A Criterion to identify maximally entangled nine-qubit state

TL;DR

This paper introduces a criterion for identifying maximally entangled nine-qubit states, proves their non-existence in absolute form, and constructs new states with specific entanglement properties.

Contribution

It establishes a generalized criterion for nine-qubit entanglement and constructs new maximally entangled states with specified balanced purities.

Findings

Absolutely maximally entangled nine-qubit states do not exist.

New nine-qubit maximally entangled states with specific balanced purities are constructed.

Marginal density matrices for subsystems are completely mixed, indicating 3-uniform states.

Abstract

We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that absolutely maximally entangled state for nine qubits does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, , 1/8 and and 1/4, respectively. We found that, the marginal density matrices for subsystems of 1,2,3- qubits all completely mixed in these states, therefore, maximally entangled nine-qubit state is a 3-uniform state.