Entanglement classification and \emph{non-k}-separability certification via Greenberger-Horne-Zeilinger-class fidelity

TL;DR

This paper introduces a practical multipartite entanglement measure that can classify and certify non-k-separability in large quantum systems using a single density matrix element, applicable to various states including GHZ, W, and Dicke states.

Contribution

It proposes a new entanglement measure that is experimentally verifiable, applicable to large systems, and capable of classifying complex multipartite entanglement structures.

Findings

Successfully classifies three-qubit SLOCC classes

Characterizes non-k-separability in four-qubit SLOCC classes

Applies to arbitrary size qubit Dicke states

Abstract

Many-body quantum systems can be characterised using the notions of \emph{k}-separability and entanglement depth. A quantum state is \emph{k}-separable if it can be expressed as a mixture of \emph{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 \emph{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, \mbox{W-,} and GHZ-type entanglement. Furthermore, we characterise the \emph{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.