Optimal estimation of three parallel spins with genuine and restricted collective measurements

Abstract: Collective measurements on identical and independent quantum systems can offer advantages in information extraction compared with individual measurements. However, little is known about the distinction between restricted collective measurements and genuine collective measurements in the multipartite setting. In this work we establish a rigorous performance gap based on a simple and old estimation problem, the estimation of a random spin state given three parallel spins. Notably, we derive an analytical formula for the maximum estimation fidelity of biseparable measurements and clarify its fidelity gap from genuine collective measurements. Moreover, we clarify the structure of optimal biseparable measurements. It turns out that the maximum estimation fidelity can be achieved by two- and one-copy measurements assisted by one-way communication in one direction, but not the other way. Our work reveals a rich landscape of multipartite nonclassicality in quantum measurements instead of quantum states and is expected to trigger further studies.