Expanding the possibility of quantum metrology with a mixture of superpositions of marcoscopically distinct states

A quantum sensor can beat the best sensitivity of a classical sensor, which is called the standard quantum limit (SQL), by the factor which scales with the square root of the system size. Sensitivity with such an enhancement is called the Heisenberg scaling. Although some specific examples that achieve the scaling have been studied, there is no general criteria about what kind of quantum states have the potential to achieve it. Here we prove that every state identified as a generalized cat state (GCS), i.e., superposition of macroscopically distinct states characterized by quantum coherence, can achieve it if used as a field sensor [1]. Importantly, even a mixture of exponentially large number of states can be identified as a GCS. We also show that GCSs beat the SQL in scaling despite the presence of dephasing which degrades the sensitivity. As a concrete example, we propose a protocol to generate a mixed GCS at finite temperature with donor spins in silicon. Its sensitivity is about 20 times better than that of a separable state even though it has an exponentially low purity.