The average determinant of the reduced density matrices for each qubit as a global entanglement measure

TL;DR

This work introduces $E_{AD}=\frac{4}{n}\sum_{i=1}^n \det \rho_i$ as a global entanglement measure for pure $n$-qubit states, linking it to the average single-qubit mixedness and the average 1-tangle. It proves a decomposition law, establishes LU invariance, and shows $E_{AD}$ coincides with Meyer–Wallach’s global entanglement measure $E_{MW}$ for all $n$, with explicit forms connecting to concurrence and $3$-tangle in low-qubit cases. The paper also relates $E_{AD}$ to von Neumann and linear entropies, showing a near-linear relation with average entropy and expressing $E_{AD}$ as the average linear entropy across qubits. It discusses the behavior of $E_{AD}$ for W and Dicke states in the large-$n$ limit, highlighting limitations for large systems and suggesting higher-order reductions to robustly capture multipartite entanglement. Overall, the results unify several entanglement concepts under a single, LU-invariant framework and clarify when $E_{AD}$ provides sharp distinctions between entanglement classes.

Abstract

In this paper, we propose the average determinant of reduced density matrices for each qubit as a global entanglement measure. By means of the properties of reduced density matrices, we can investigate the present measure. We propose a decomposition law for the present measure, demonstrate that the present measure just measures the average mixedness for each qubit and the average 1-tangle, and indicate that for n-qubit W state and n-qubit Dicke states, the average mixedness for each qubit and 1-tangle almost vanish for the large number of qubits. We also point out that for two quits, the present measure is just the square of the concurrence while for three qubits, the present measure is the sum of 3-tangle and the double of the average 2-tangles.