Quantum metrology of rotations with mixed spin states

The efficiency of a quantum metrology protocol can be significantly diminished by the interaction of the system with its environment, leading to a loss of purity and, as a result, a mixed state for the probing system. An example is the measurement of a magnetic field through the rotation of a spin that is subject to decoherence due to its coupling to a surrounding spin or bosonic bath. In this work we define mixed optimal quantum rotosensors (OQRs) as mixed spin-$j$ states that achieve maximum sensitivity to estimate infinitesimal rotations, when the rotation axis is unknown. We study two scenarios, where the probe states saturate the averaged fidelity or the averaged quantum Cram\'er-Rao bound, the latter giving the ultimate sensitivity. We find that mixed OQRs can achieve sensitivity equal to that of pure states and are obtained by mixing states from linear subspaces of anticoherent states. We present several examples of mixed OQRs and their associated anticoherent subspaces. We also show that OQRs maximize entanglement in a specific sense, preserving the known relation between entanglement and optimal rotation sensitivity for pure states, even in the context of mixed states. Our results highlight the interconnection between quantum metrology of rotations, anticoherence, and entanglement in mixed spin states.