Entanglement detection via quantum Fisher information

We propose an entanglement criterion in terms of quantum Fisher information, which complements the criterion based on variance and local uncertainty relations [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. We illustrate the significance of this criterion by showing that it can reveal entanglement undetectable by the variance method. The dual relation between the criterion based on quantum Fisher information and that based on variance are highlighted. The combination of these two criteria leads to a refined method for entanglement detection, which is stronger than when either one is used alone.

Figure 1

Variance $V_p={\sum }_{\mu }V({\rho }_p^{ab},A_{\mu }\otimes {\mathbf{1}}^b+{\mathbf{1}}^a\otimes B_{\mu })$ and quantum Fisher information $F_p={\sum }_{\mu }F({\rho }_p^{ab},A_{\mu }\otimes {\mathbf{1}}^b+{\mathbf{1}}^a\otimes B_{\mu })$ versus the parameter $p$ in the state ${\rho }_p^{ab}$, as defined by Eq. (12). We see clearly that in this case, the entanglement criterion based on variance is inconclusive in detecting entanglement of ${\rho }_p^{ab}$ since inequality (9) is not violated for any $p$; while by the entanglement criterion based on quantum Fisher information, we conclude that ${\rho }_p^{ab}$ is entangled whenever $p>2/3$ since inequality (8) is violated for such a $p$.