Abstract
We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.
Highlights
Capability of performing precise measurements is the cornerstone of modern physics
Most of the bounds derived in the field of quantum metrology, including t√he ones mentioned above, are applications of the celebrated Quantum Cramer-Rao (C-R) bound [3] ∆φ ≥ 1/ kF which is based on calculation of the Quantum Fisher Information (QFI)
F = tr ρφL2φ, where k is the number of independent repetitions of experiment, ρφ = Λφ(|ψN ψN |) is the output state of the channel Λφ which imprints value φ of the parameter we want to estimate on the input pure state |ψN of N probes and Lφ is an operator called symmetric logarithmic derivative (SLD) given by equation dρφ dφ
Summary
Capability of performing precise measurements is the cornerstone of modern physics. Quantum mechanics provides insight into fundamental limits on the achievable measurement precision that cannot be beaten irrespectively of the extent of any improvements in measurement technology. Saturating the C-R bound may require unrealistically good prior knowledge on the value of the estimated parameter.
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