to what quantumness is employed for improving the performance of measurements: (I) quantum object such as electrons or nuclear spins, (II) quantum coherence such as superposition states or matter-wave-like nature, and (III) quantum entanglement that cannot be described classically [@RevModPhys.89.035002]. The third may be the most quantum-like one, and various efforts have been reported along this direction. The entanglement-enhanced magnetic field sensing with atomic vapors were reported: the spin squeezing (entanglement) within the atomic internal structure [@PhysRevLett.101.073601], and the entanglement between two vapor cells [@PhysRevLett.104.133601] were employed for reducing noise. More direct approach enhancing the sensitivity of measurements with entanglement was discussed [@Giovannetti:2011aa] and experimental efforts by using trapped ions [@Leibfried:2003aa; @Leibfried1476; @Leibfried:2005aa; @PhysRevLett.86.5870], ultra cold atoms [@Orzel2386; @Appel10960], and NMR [@Jones1166; @PhysRevA.82.022330] were reported.

A potential problem of the entangled sensor is the fragility against noise. In fact, it is theoretically pointed out that an entangled sensor in a Markovian noisy environment, where a relaxation is exponential, cannot overcome the standard quantum limit (SQL) [@huelga1997improvement]. The SQL is the consequence of the central-limit theorem [@Giovannetti:2011aa]. On the other hand, there is a theoretical prediction that the entangled sensor can outperform the classical sensor under the effect of a time-inhomogeneous 