Entanglement classification and non-k-separability certification via Greenberger-Horne-Zeilinger-class fidelity

Many-body quantum systems can be characterised 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 GHZ-type entanglement. Furthermore, we characterise the non-k-separability in known nine SLOCC classes of four-qubit states, absolutely maximally entangled states for five and six qubits and for arbitrary size qubit Dicke states.