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arXiv:2301.07380 · Physical Review A

Protocol for global multiphase estimation

Giovanni Chesi, Alberto Riccardi, Roberto Rubboli, Lorenzo Maccone, Chiara Macchiavello

Published Jul 17, 2023• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.108.012613

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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.
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.
Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, applications to the simultaneous estimation of several phases may bring substantial advantages; for instance, in the presence of spatial or temporal constraints. In this work, we study a Bayesian algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The protocol gives access to correlations in the Bayesian multiphase distribution resulting in covariance matrix elements scaling as $O({N}_{T}^{\ensuremath{-}2})$, with respect to the total number of quantum resources NT{N}_{T}NT​. The parallel estimation allows to surpass the sensitivity of sequential single-phase estimation strategies for optimal linear combinations of phases. Furthermore, the algorithm proves a certain noise resilience and can be implemented using single photons and standard optical elements in currently accessible experiments.
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.
Quantum phase estimation (QPE) is ubiquitous in quantum technologies. It is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, one can envisage applications where the simultaneous estimation of multiple phases may bring substantial advantages, for instance in presence of spatial or temporal constraints. In this work, we study a Bayesian QPE algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The algorithm allows the estimation of each phase, as well as any arbitrary linear combination, with Heisenberg-limited sensitivity with respect to the total number of quantum resources. The parallel estimation protocol gives access to correlations in the Bayesian multiphase distribution that allow to surpass the sensitivity of sequential single phase estimation strategies for optimal combinations of phases. Furthermore, the algorithm proves a certain noise resilience, and can be implemented using single photons and standard optical elements in currently accessible experiments.
Quantum phase estimation is fundamental to advancing quantum science and technology. While much of the research has concentrated on estimating a single phase, the simultaneous estimation of multiple phases can yield significantly enhanced sensitivities when using specially tailored input quantum states. This work reviews recent theoretical and experimental advancements in the parallel estimation of multiple arbitrary phases. We highlight strategies for constructing optimal measurement protocols and discuss the experimental platforms best suited for implementing these techniques.
We present a general quantum metrology framework to study the simultaneous estimation of multiple phases in the presence of noise as a discretized model for phase imaging. This approach can lead to nontrivial bounds of the precision for multiphase estimation. Our results show that simultaneous estimation (SE) of multiple phases is always better than individual estimation (IE) of each phase even in noisy environment. The utility of the bounds of multiple phase estimation for photon loss channels is exemplified explicitly. When noise is low, those bounds possess the Heisenberg scale showing quantum-enhanced precision with the O(d) advantage for SE, where d is the number of phases. However, this O(d) advantage of SE scheme in the variance of the estimation may disappear asymptotically when photon loss becomes significant and then only a constant advantage over that of IE scheme demonstrates. Potential application of those results is presented.