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arXiv:1703.00063 · Physical Review A

Quantum multiparameter estimation with generalized balanced multimode NOON-like states

Lu Zhang, Kam Wai Clifford Chan

Published Mar 20, 2017• 54 citations
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Abstract

The simultaneous multiparameter estimation problem using a class of multimode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered, and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multimode NOON state. The analytical form for the quantum Cram\'er-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. Finally, we illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios.

DOI
10.1103/physreva.95.032321

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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•May 1, 2011•1 citation
The simultaneous multiparameter estimation problem using a class of multimode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered, and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multimode NOON state. The analytical form for the quantum Cram'er-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. Finally, we illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios.
Quantum multiple parameter estimation can achieve an enhanced sensitivity beyond the classical limit. Although a theoretical ultimate sensitivity bound for multiple phase estimation is given by the quantum Cramer-Rao bound (QCRB), experimental implementations to saturate the QCRB typically require an impractical setup including entangled measurements. Since it is experimentally challenging to implement an entangled measurement, the practical sensitivity is given by the Cramer-Rao bound (CRB) relevant to the measurement probabilities. Here, we consider the problem of practical sensitivity bound for multiple phase estimation with quantum probe states and a measurement setup without entanglement, which consists of a beam splitter followed by the photon-number-resolving measurement. In this practical measurement scheme, we show that lower CRB can be achieved with a quantum probe state even with higher QCRB.
Abstract We propose a generalized form of entangled coherent states (ECS) and apply them in a multi-arm optical interferometer to estimate multiple phase shifts. We obtain the quantum Cramér–Rao bounds for both the linear and nonlinear parametrization protocols. Through analysis utilizing simultaneous estimation we find that this generalized form of ECS gives better precision than the generalized NOON states (2013 Phys. Rev. Lett. 111 070403 ). Moreover, when compared with the independent estimation both the linear and nonlinear protocols have the same advantage in the relation to the number of parameters.
This paper explores multiparameter quantum metrology using Greenberger–Horne–Zeilinger (GHZ)-type photon-added coherent states (PACS) and investigates both independent and simultaneous parameter estimation with linear and nonlinear protocols, highlighting the significant potential of quantum resources to enhance precision in multiparameter scenarios. To provide a comprehensive analysis, we explicitly derive analytical expressions for the quantum Cramér–Rao bound (QCRB) for each protocol. Additionally, we compare the two estimation strategies, examining the behavior of their QCRBs and offering insights into the advantages and limitations of these quantum states in various contexts. Our results show that simultaneous estimation generally outperforms independent estimation, particularly in nonlinear protocols. Furthermore, we analyze how the QCRB varies with the coherent state amplitude [Formula: see text], the number of estimated parameters d, and the photon excitation order n across three protocols. The results indicate that increasing [Formula: see text] and decreasing d improves estimation precision. For low n, the variation in the QCRB is similar for both symmetric and antisymmetric cases; however, at higher n, the antisymmetric case exhibits slightly better precision. The dependence on d is comparable for both types of states. We also compare PACS-based GHZ states with NOON states and entangled coherent states, demonstrating the relative performance of each. Finally, we conclude with an analysis of homodyne detection in the context of a linear protocol, discussing its impact on estimation accuracy.
The multiparameter estimation in quantum metrology is significant for a large number of applications for quantum information tasks. Nevertheless, how mode correlations affect the multiparameter-estimation precision with and without photon-loss cases remains ambiguous. For this reason, here, we present the relationship between the quantum Cram\'er-Rao bound and the mode correlations in the framework of the simultaneous two-phase estimation for a three-mode Mach-Zehnder interferometer induced by the tritter which is composed of three cascaded beam splitters and phase shifters. By choosing different input resources, including a two-mode squeezed vacuum state $\ensuremath{\bigotimes}$ a coherent state (scheme I) and a twin squeezed vacuum state $\ensuremath{\bigotimes}$ a coherent state (scheme II), the results show that compared with scheme II, scheme I shows a better estimation performance even in the presence of photon losses. The reason may be that the intramode correlation of the probe state for scheme I is stronger than that for scheme II, while the intermode correlation of the probe state for the former is weaker than that for the latter. This allow us to open vital perspectives on the essence of enhanced multiparameter-estimation precision.
Multimode Gaussian quantum light, which includes coherent light, but also all kinds of multimode squeezed and/or multipartite quadrature entangled states, is a very general and powerful quantum resource that is now currently produced in various laboratories in the world, and is successfully used in various applications to quantum information processing[1]. In contrast to non-classical states based on the manipulation of Fock states (such as the NOON states), such quantum light can be produced with very high values of the mean photon number N (up to 1016). It is therefore a very good candidate for quantum metrology purposes, i.e. to improve as much as possible the estimation of a parameter p that is encoded on the spatio-temporal variation of a light beam, as all the limits in parameter estimation scale with some inverse power of N.
Multimode Gaussian quantum light, which includes coherent light, but also all kinds of multimode squeezed and/or multipartite quadrature entangled states, is a very general and powerful quantum resource that is now currently produced in various laboratories in the world, and is successfully used in various applications to quantum information processing. In contrast to non-classical states based on the manipulation of Fock states (such as the NOON states), such quantum light can be produced with very high values of the mean photon number N (up to 1016). It is therefore a very good candidate for quantum metrology purposes, i.e. to improve as much as possible the estimation of a parameter p that is encoded on the spatio-temporal variation of a light beam, as all the limits in parameter estimation scale with some inverse power of N.