Abstract
Quantum metrology offers enhanced performance in experiments on topics such as gravitational wave-detection, magnetometry or atomic clock frequency calibration. The enhancement, however, requires a delicate tuning of relevant quantum features, such as entanglement or squeezing. For any practical application, the inevitable impact of decoherence needs to be taken into account in order to correctly quantify the ultimate attainable gain in precision. We compare the applicability and the effectiveness of various methods of calculating the ultimate precision bounds resulting from the presence of decoherence. This allows us to place a number of seemingly unrelated concepts into a common framework and arrive at an explicit hierarchy of quantum metrological methods in terms of the tightness of the bounds they provide. In particular, we show a way to extend the techniques originally proposed in Demkowicz-Dobrzański et al (2012 Nature Commun. 3 1063), so that they can be efficiently applied not only in the asymptotic but also in the finite number of particles regime. As a result, we obtain a simple and direct method, yielding bounds that interpolate between the quantum enhanced scaling characteristic for a small number of particles and the asymptotic regime, where quantum enhancement amounts to a constant factor improvement. Methods are applied to numerous models, including noisy phase and frequency estimation, as well as the estimation of the decoherence strength itself.
Highlights
Quantum enhanced metrology has recently enjoyed a great success at experimental level leading to new results in atomic spectroscopy [1,2,3,4], magnetometry [5,6,7] and optical interferometry [8,9,10,11] with prominent achievements in gravitational waves sensing [12]
An important question that has been considerably addressed by many researchers [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] reads: how and to what extent can the noise effects be compensated in quantum metrological setups? In the case of atomic spectroscopy, it has already been indicated in [24, 25] that the effects of uncorrelated noise, independently affecting the atoms within a probe, have a dramatic impact on quantum protocols—most likely restricting the ultimate precision scaling to become Standard Quantum Limit (SQL)-like for high enough N, so that the quantum enhancement is asymptotically limited to a multiplicative constant factor
As in metrological setups the estimated parameter is encoded in the evolution of a system, we identify ρφ = Λφ[ρin] as the final state of a system that started from an input ρin
Summary
Quantum enhanced metrology has recently enjoyed a great success at experimental level leading to new results in atomic spectroscopy [1,2,3,4], magnetometry [5,6,7] and optical interferometry [8,9,10,11] with prominent achievements in gravitational waves sensing [12]. The uncertainty in reconstructing the encoded parameter, such as e.g. optical phase delay or frequency difference, can√in principle be proportional to 1/N, the so-called Heisenberg Limit (HL), rather than 1/ N , commonly referred to as the Standard Quantum Limit (SQL) or the shot (projection) noise This dramatic scaling improvement can be illusive, as both the experimental results and theoretical toy-models have indicated that achieving the ideal HL is a hard task owing to the strong destructive impact of imperfections, which should be always accounted for in realistic scenarios. We apply our results to phase/frequency estimation with various noise models including: dephasing, depolarization, loss and spontaneous emission, restricting ourselves to the cases of noise commuting with the parameterized unitary part of the evolution This assumption makes the analysis more transparent, but is not indispensable, since our methods may be effectively employed for any single particle evolution described by a general Lindblad equation [54] reshaped into the corresponding Kraus representation [55].
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