Detecting $k$-nonseparability via quantum Fisher information

The $k$-nonseparability is a kind of multipartite quantum correlation which can be used to characterize some aspects of multipartite entanglement. In this paper, we discuss the detection of $k$-nonseparability of $n$-partite quantum states from an information perspective and obtain some $k$-nonseparability criteria in terms of quantum Fisher information. These criteria can identify many $k$-nonseparable states, and their advantages over an important and well-known $k$-nonseparability criterion [M. Huber et al., Phys. Rev. A 88, 042328 (2013)] are illustrated by several typical examples.

Figure 1

Comparison of the detection power between Proposition 2 and inequality (12) for $\rho (p,q)=(1-p){\rho }_{\text{GHZ}}+q{\rho }_{\text{dep}}+\frac{p-q}{2^{10}}\mathbf{1}$ when $k=7$ and $k=8$. The area enclosed by line $a$, the $p$ axis, line $q=p$, and line $q=1-p$ corresponds to the 7-nonseparable states detected by Proposition 2; the area enclosed by line $a^{\prime }$, the $p$ axis, line $q=p$, and line $q=1-p$ corresponds to the 7-nonseparable states detected by inequality (12). Clearly, the 7-nonseparable states in the area enclosed by line $a$, the $p$ axis, line $a^{\prime }$, and line $q=1-p$ are detected only by Proposition 2 and not by inequality (12). Similarly, the 8-nonseparable states in the area enclosed by line $b$, the $p$ axis, line $b^{\prime }$, and line $q=1-p$ are detected only by Proposition 2 and not by inequality (12).