Potential multiparticle entanglement measure

Abstract

In this Brief Report we discuss entanglement of multiparticle quantum systems. We propose a potential measure of a type of entanglement of pure states of $n$ qubits, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}.$ For a system of two qubits the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is equal to the square of the concurrence, and for systems of three qubits it is equal to the ``residual entanglement.'' We show that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is also equal to a generalization of the concurrence squared for even $n,$ and use this fact to prove that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is an entanglement monotone. However, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is undefined for odd $n>3.$ Finally, we propose a measure related to the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ for mixed-state systems of $n$ qubits, and find an analytical formula for this measure for even $n.$