Optimal quantum precision in noise estimation: Necessity of entanglement

We ask whether the optimal probe is entangled, and if so, what is its character and amount, for estimating the noise parameter of a large class of local quantum encoding processes that we refer to as vector encoding, examples of which include the local depolarizing and bit-flip channels. We first establish that vector encoding is invariably ``continuously commutative'' for optimal probes. We utilize this result to deal with the queries about entanglement in the optimal probe. We show that for estimating noise extent of the two-party arbitrary-dimensional local depolarizing channel, there is a descending staircase of optimal-probe entanglement for increasing depolarizing strength. For the multi-qubit case, the analysis again leads to a staircase, but which can now be monotonic or not, depending on the multiparty entanglement measure used. We also find that when sufficiently high depolarizing noise is to be estimated, fully product multiparty states are the only choice for being optimal probes. In many cases, for even moderately high depolarizing noise, fully product states are optimal. For two-qubit local bit-flip channels, the continuous commutativity of the channel and optimal probe implies that a product state suffices for obtaining the optimal precision.