Experimentally feasible set of criteria detecting genuine multipartite entanglement in $n$-qubit Dicke states and in higher-dimensional systems

We construct a set of criteria detecting genuine multipartite entanglement in arbitrary dimensional multipartite systems. These criteria are optimally suited for detecting multipartite entanglement in $n$-qubit Dicke states with $m$ excitations, as shown in exemplary cases. Furthermore, they can be employed to detect multipartite entanglement in different states related to quantum cloning, decoherence-free communication, and quantum secret sharing. In a detailed analysis, we show that the criteria are also more robust to noise than any other criterion known so far, especially with increasing system size. Furthermore, it is shown that the number of required local observables scales only polynomially with size, thus making the criteria experimentally feasible.

Figure 1

This plot shows the detection quality of Eq. (5) for the six-qubit state $\rho =p{\rho }_{D_2}+q{\rho }_{D_3}+\frac{1-p-q}{2^6}\mathbb{1}$, where ${\rho }_{D_2}=|D_2^6\rangle \langle D_2^6|$ and ${\rho }_{D_3}=|D_3^6\rangle \langle D_3^6|$. Region $I:D_2$ represents the parameter region for which the state is detected to be genuinely multipartite entangled by Eq. (5) corresponding to $|D_2^6\rangle$ and, analogously, $I:D_3$ corresponds in the same way to Eq. (5) using $|D_3^6\rangle$. The region $F:D_3$ depicts the parameter region detected by the fidelity witness used in Ref. [33], which demonstrates the improvement provided by Eq. (5).

Figure 2

The plot shows the white noise resistance parameter $p$ in terms of the number of qubits $n$, where, from left to right, the resistance is plotted for $m=2$ to 20 in ascending order.