Entanglement classification and non-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-separability certification via Greenberger-Horne-Zeilinger-class fidelity

Many-body quantum systems can be characterized using the notions of $k$-separability and entanglement depth. A quantum state is $k$-separable if it can be expressed as a mixture of $k$ entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-$k$-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, $W$-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-$k$-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.