Tolerable state-preparation costs for phase-shift parameter estimation with very noisy states

We consider protocols for estimating a single qubit phase shift parameter when the available initial states are mixed. Using the quantum Fisher information (QFI) to quantify extimation accuracy, we compare protocols using a single channel query on n qubits to a single channel query on one qubit. A well-established existing protocol shows that the n-qubit protocol using a particular correlated state prepared prior to the channel query can enhance the accuracy. For very noisy initial states the correlated state protocol increases the QFI by a factor of n.

We now extend this to consider the cost of preparing the correlated state. We quantify the preparation cost in terms of its required two-qubit unitaries and provide an expression that shows a trade-off in QFI gains versus the preparatory unitary cost. We show that reducing the cost of the preparatory unitaries reduces QFI gains, but that there can still be useful gains. We explore this when the initial states are very noisy and show that even with less costly preparatory unitaries, it will always be advantageous to increase the number of qubits used.