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arXiv:2006.13230 · Physical Review A

Multiphase estimation without a reference mode

Aaron Z. Goldberg, Ilaria Gianani, Marco Barbieri, Fabio Sciarrino, Aephraim M. Steinberg, Nicolò Spagnolo

Published Aug 27, 2020• 28 citations

Abstract

Multiphase estimation is a paradigmatic example of a multiparameter problem. When measuring multiple phases embedded in interferometric networks, specially tailored input quantum states achieve enhanced sensitivities compared with both single-parameter and classical estimation schemes. Significant attention has been devoted to defining the optimal strategies for the scenario in which all of the phases are evaluated with respect to a common reference mode, in terms of optimal probe states and optimal measurement operators. Also, the strategies assume unlimited external resources, which is experimentally unrealistic. Here, we optimize a generalized scenario that treats all of the phases on an equal footing and takes into account the resources provided by external references. We show that the absence of an external reference mode reduces the number of simultaneously estimable parameters, owing to the immeasurability of global phases, and that the symmetries of the parameters being estimated dictate the symmetries of the optimal probe states. Finally, we provide insight for constructing optimal measurements in this generalized scenario. The experimental viability of this paper underlies its immediate practical importance beyond fundamental physics.

DOI: 10.1103/physreva.102.022230

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We consider the problem of estimating multiple phases using a multimode interferometer. In this setting we show that while global strategies that estimate all the phases simultaneously can lead to high precision gains, the same enhancements can be obtained with local strategies in which each phase is estimated individually. A key resource for the enhancement is shown to be a large particle-number variance in the probe state, and for states where the total particle number is not fixed, this can be obtained for mode-separable states, and the phases can be read out with local measurements. This has important practical implications because local strategies are generally preferred to global ones for their robustness to local estimation failure, flexibility in the distribution of resources, and comparatively easier state preparation. We obtain our results by analyzing two different schemes: the first uses a set of interferometers, which can be used as a model for a network of quantum sensors, and the second looks at measuring a number of phases relative to a reference, which is concerned primarily with quantum imaging.

In this work, we investigate the problem of multiphase estimation using generalized

3

- and

4

-mode Mach-Zehnder interferometer. In our setup, we assume that the number of unknown phases is the same as the number of modes in the interferometer, which introduces strong correlations between estimators of the phases. We show that despite these correlations and despite the lack of optimisation of a measurement strategy (a fixed interferometer is used) we can still obtain the Heisenberg-like scaling of precision of estimation of all the parameters. Our estimation scheme can be applied to the task of quantum-enhanced sensing in 3-dimensional interferometric configurations.

Abstract Quantum multiparameter metrology is hindered by incompatibility issues, such as finding a single probe state (probe incompatibility) and a single measurement (measurement incompatibility) optimal for all parameters. The simultaneous estimation of phase shift and loss in a single optical mode is a paradigmatic multiparameter metrological problem in which such tradeoffs are present. We consider two settings: single-mode or two-mode probes (with a reference lossless mode), and for each setting we consider either Gaussian states or arbitrary quantum states of light restricted only by a maximal number of photons allowed. We find numerically that, as the number of photons increases, there are quantum states of light for which probe incompatibility disappears both in the single- and two-mode scenarios. On the other hand, for Gaussian states, probe incompatibility is present in the single-mode case and may be removed only in the two-mode setting thanks to the entanglement with the reference mode. Finally, we provide strong arguments that the fundamental incompatibility aspect of the model is measurement incompatibility, which persists for all the scenarios considered, and unlike probe-incompatibility cannot be overcome even in the large photon number limit.

Global estimation strategies allow one to extract information on a phase or a set of phases without any prior knowledge, which is required for local estimation strategies. We devise a global multiphase protocol based on Holevo's estimation theory and apply it to the case of digital estimation, i.e., we estimate the phases in terms of the mutual information between them and the corresponding estimators. We test the protocol in two particular cases, i.e., the single-phase and the double-phase estimation. In the single-phase scenario, the protocol encompasses two specific known optimal strategies. We extend them to the simultaneous estimation of two phases and evaluate their performance. Then we retrieve the ultimate digital bound on precision when a generic number of phases is simultaneously estimated. This bound is again expressed in terms of mutual information and is general for any digital multiphase estimation protocol. We show that in the multiphase strategy there is only a constant quantum advantage with respect to a sequence of independent single-phase estimations. This extends a recent similar result, which settled a controversy on the search for the multiphase enhancement.

A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this Letter, we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the ultimate theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.