Noise-dependent optimal strategies for quantum metrology

For phase estimation using qubits, we show that for some noise channels, the optimal entanglement-assisted strategy depends on the noise level. We note that there is a nontrivial crossover between the parallel-entangled strategy and the ancilla-assisted strategy: in the former the probes are all entangled; in the latter the probes are entangled with a noiseless ancilla but not among themselves. The transition can be explained by the fact that separable states are more robust against noise and therefore are optimal in the high-noise limit, but they are in turn outperformed by ancilla-assisted ones.