Optimal extraction of information from two spins

We revisit the issue of extracting information from parallel and antiparallel spins, as initiated by Gisin and Popescu [Phys. Rev. Lett. 83, 432 (1999)], from the viewpoint of parameter estimation. By comparing two fundamental figures of merit based on fidelity and quantum Fisher information for assessing information quality, we demonstrate that these criteria yield different strategies for optimally extracting spin information. A surprising observation is that while for a single spin, as well as for parallel spins, quantum Fisher information cannot be fully extracted by any uniform measurement, this is not the case for antiparallel spins. A simple uniform measurement fully extracting quantum Fisher information in antiparallel spins is identified. This reveals a significant feature of antiparallel spins from the perspective of Fisher information and provides an alternative illustration of the idea that antiparallel spins carry more information than parallel spins.

Figure 1

Quantum Fisher information $Q_{\theta }(\Pi \left({\rho }_{\parallel }\right))$ and $Q_{\varphi }(\Pi \left({\rho }_{\parallel }\right))$ as functions of $\theta \in [0,\pi ),\varphi \in [0,2\pi ).$ This should be compared with the quantum Fisher information $Q_{\theta }(\Phi \left({\rho }_{\parallel }\right))=1$ and $Q_{\varphi }(\Phi \left({\rho }_{\parallel }\right))={\sin }^2\theta .$ Their average values (with respect to the uniform probability measure on the Bloch sphere) have the following relations: ${\overline{Q}}_{\theta }(\Phi \left({\rho }_{\parallel }\right))=1<{\overline{Q}}_{\theta }(\Pi \left({\rho }_{\parallel }\right))\simeq 1.1357,$ ${\overline{Q}}_{\varphi }(\Phi \left({\rho }_{\parallel }\right))=\frac{2}{3}={\overline{Q}}_{\varphi }(\Pi \left({\rho }_{\parallel }\right)).$ The measurement $\Pi$ performs better than the measurement $\Phi$ in extracting Fisher information from parallel spins.

Figure 2

Quantum Fisher information $Q_{\theta }(\Phi \left({\rho }_{\perp }\right))$ and $Q_{\varphi }(\Phi \left({\rho }_{\perp }\right))$ as functions of $\theta \in [0,\pi ),\varphi \in [0,2\pi ).$ This should be compared with the quantum Fisher information $Q_{\theta }(\Pi \left({\rho }_{\perp }\right))=2$ and $Q_{\varphi }(\Pi \left({\rho }_{\perp }\right))=2{\sin }^2\theta .$ Their average values (with respect to the uniform probability measure on the Bloch sphere) have the following relations: ${\overline{Q}}_{\theta }(\Phi \left({\rho }_{\perp }\right))\simeq 0.9025<{\overline{Q}}_{\theta }(\Pi \left({\rho }_{\perp }\right))=2,$${\overline{Q}}_{\varphi }(\Phi \left({\rho }_{\perp }\right))\simeq 0.6026<{\overline{Q}}_{\varphi }(\Pi \left({\rho }_{\perp }\right))=\frac{4}{3}.$ The measurement $\Pi$ performs better than the measurement $\Phi$ in extracting Fisher information from antiparallel spins.