Qubit-channel metrology with highly noisy initial states and additional undesirable noisy evolution

We consider protocols for estimating the parameter in a single-parameter unital qubit channel, assuming that the available initial states are highly mixed with very low purity. We compare two protocols, each querying the channel once, via their quantum Fisher informations. One uses $n$ qubits prepared in a particular correlated input state and subsequently queries the channel on one qubit. The other uses a single qubit. We extend the results of Collins [Phys. Rev. A 99, 012123 (2019)] by allowing for additional noisy evolution on the spectator qubits in the $n$-qubit protocol. We provide simple algebraic expressions that will determine when the $n$-qubit protocol is superior. We provide a technique that can alleviate certain types of noise. We show that for certain types of noisy evolution the $n$-qubit protocol will be inferior but for others it will be superior.