Optimal measurements for relative quantum information

We provide optimal measurement schemes for estimating relative parameters of the quantum state of a pair of spin systems. We prove that the optimal measurements are joint measurements on the pair of systems, meaning that they cannot be achieved by local operations and classical communication. We also demonstrate that in the limit where one of the spins becomes macroscopic, our results reproduce those that are obtained by treating that spin as a classical reference direction.

Figure 1

Average information gain for measurements on a spin-$1∕2$ system and a spin-$j$ system. The curves (a),(b) correspond to the optimal measurement and the optimal local measurement for the case when the spins are prepared parallel or antiparallel with equal probability. The curves (c),(d) correspond to the optimal measurement and the optimal local measurements for the case when the initial direction of each spin is chosen uniformly from the sphere.