Bounding the benefit of adaptivity in quantum metrology using the relative fidelity

TL;DR

This paper introduces the relative fidelity to bound the performance of adaptive quantum channel discrimination and estimation protocols, providing a way to quantify the benefit of adaptivity and entanglement.

Contribution

It defines the relative fidelity as a new metric to bound adaptive protocol performance in quantum metrology, linking it to output state fidelities and quantum Fisher information.

Findings

Bound on output state fidelity in terms of relative fidelity

Performance limits for adaptive quantum discrimination and estimation

Quantum Fisher information constrained by protocol size

Abstract

Protocols for discriminating between a pair of channels or for estimating a channel parameter can often be aided by adaptivity or by entanglement between the probe states. This can make it difficult to bound the best possible performance for such protocols. In this paper, we introduce a quantity that we call the relative fidelity of a given pair of channels and a pair of input states to those channels. Constraining the allowed input states to all pairs of states whose fidelity is greater than some minimum "input fidelity" and minimising this quantity over the valid pairs of states, we get the minimum relative fidelity for that input fidelity constraint. We are then able to lower bound the fidelity between the possible output states of any protocol acting on one of two possible channels in terms of the minimum relative fidelity. This allows us to bound the performance of the most general, adaptive discrimination and parameter estimation protocols. By finding a continuity bound for the relative fidelity, we also provide a simple confirmation that the quantum Fisher information (QFI) of the output of an $N$-use protocol is no more than $N^2$ times the one-shot QFI.