Fig. 1.

Schematic sketch of the metrological task. An agent holds a physical system of spin $J=3/2$, in an initial state $\rho$. She gives it to a referee, who, in a black box with respect to the agent, performs some spatial rotation of angle $\theta$ on the system, where $\theta$ is chosen according to the distribution function $\mu (\theta )$ (defined in the main text and shown in Fig. 2). The referee then passes the system back to the agent, who measures it using a two-outcome box in order to determine whether the angle $\theta$ is in the range $R_{+}$ or $R_{-}$ (see also Fig. 2)