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Journal of Physics A Mathematical and General

Phase estimation in interferometry

Michael Zawisky, Yuji Hasegawa, H. Rauch, Z. Hradil, Robert Myška, Jan Peřina

Published Jan 16, 1998• 29 citations
Publisher

Abstract

An objective theory of phase shift estimation is formulated within quantum mechanics and quantum estimation theory. The general validity of Bayesian analysis even in the limit of very low particle numbers is demonstrated using the information of measured phase-sensitive data. The formalism was applied to the estimation of an unknown phase in neutron interferometry operating near the quantum limit, when only a few particles are registered, and also in the regime of large particle numbers.

DOI
10.1088/0305-4470/31/2/015

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An objective theory of phase shift estimation is formulated within quantum mechanics and quantum estimation theory. The general validity of Bayesian analysis even in the limit of very low particle numbers is demonstrated using the information of measured phase-sensitive data. The formalism was applied to the estimation of an unknown phase in neutron interferometry operating near the quantum limit, when only a few particles are registered, and also in the regime of large particle numbers.
Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic 1/n1/n1/n decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for nnn the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the 1/n1/n1/n limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.
Phase shift estimation with uncertainty below the Heisenberg limit, $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}$, where N‾T{\overline{N}}_{T}NT​ is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty $\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}$, with kkk an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}$ at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.
Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér-Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér-Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.
Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.
Abstract Realisation of experiments even on small and medium-scale quantum computers requires an optimisation of several parameters to achieve high-fidelity operations. As the size of the quantum register increases, the characterisation of quantum states becomes more difficult since the number of parameters to be measured grows as well and finding efficient observables in order to estimate the parameters of the model becomes a crucial task. Here we propose a method relying on application of Bayesian inference that can be used to determine systematic, unknown phase shifts of multi-qubit states. This method offers important advantages as compared to Ramsey-type protocols. First, application of Bayesian inference allows the selection of an adaptive basis for the measurements which yields the optimal amount of information about the phase shifts of the state. Secondly, this method can process the outcomes of different observables at the same time. This leads to a substantial decrease in the resources needed for the estimation of phases, speeding up the state characterisation and optimisation in experimental implementations. The proposed Bayesian inference method can be applied in various physical platforms that are currently used as quantum processors.