Adversarial versus cooperative quantum estimation

We address the estimation of a one-parameter family of isometries taking one input into two output systems. This primarily allows us to consider imperfect estimation by accessing only one output system, i.e., through a quantum channel. Then, on the one hand, we consider separate and adversarial control of the two output systems to introduce the concept of privacy of estimation. On the other hand we conceive the possibility of separate but cooperative control of the two output systems. Optimal estimation strategies are found according to the minimum mean square error. This also implies the generalization of Personik’s theorem to the case of local measurements. Finally, applications to two-qubit unitaries (with one qubit in a fixed input state) are discussed.