Relationship between the n-tangle and the residual entanglement of even n qubits

TL;DR

This paper explores the relationship between the n-tangle and residual entanglement in even n-qubit systems, revealing that n-tangle is not a residual entanglement measure and analyzing their properties and differences.

Contribution

It establishes that n-tangle is the square of a degree-2 SLOCC polynomial invariant and clarifies its limitations as an entanglement measure for even n-qubits.

Findings

n-tangle is the square of a degree-2 SLOCC polynomial invariant

n-tangle is not the residual entanglement for even n≥4 qubits

Concurrence C_{1(2...n)} is always positive for entangled states, unlike n-tangle

Abstract

We show that $n$-tangle, the generalization of the 3-tangle to even $n$ qubits, is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$-tangle is not the residual entanglement for any even $n\geq 4$\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$. The condition implies that the concurrence $% C_{1(2...n)}$ is always positive for any entangled states while the $n$% -tangle vanishes for some entangled states. We argue that for even $n$\ qubits, the concurrence $C_{1(2...n)}$\ is equal to or greater than the $n$% -tangle. Further,\ we reveal that the residual entanglement is a partial measure for product states of any $n$ qubits while the $n$-tangle is multiplicative for some product states.