Relations among k-ME concurrence, negativity, polynomial invariants, and tangle

The k-ME concurrence as a measure of multipartite entanglement (ME) unambiguously detects all k-nonseparable states in arbitrary dimensions and satisfies many important properties of an entanglement measure. Negativity is a simple computable bipartite entanglement measure. Invariant and tangle are useful tools to study the properties of the quantum states. In this paper, we mainly investigate the internal relations among the k-ME concurrence, negativity, polynomial invariants, and tangle. Strong links between k-ME concurrence and negativity as well as between k-ME concurrence and polynomial invariants are derived. We obtain the quantitative relation between k-ME \((k=n)\) concurrence and negativity for all n-qubit states, give an exact value of the n-ME concurrence for the mixture of n-qubit GHZ states and white noise, and derive an connection between k-ME concurrence and tangle for n-qubit W state. Moreover, we find that for any 3-qubit pure state the k-ME concurrence \((k=2, 3)\) is related to negativity, tangle, and polynomial invariants, while for 4-qubit states the relations between k-ME concurrence (for \(k = 2, 4)\) and negativity, and between k-ME concurrence and polynomial invariants also exist. Our work provides clear quantitative connections between k-ME concurrence and negativity, and between k-ME concurrence and polynomial invariants.