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 a 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.

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