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.
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.
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.
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.
Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision (N is the mean photon number of the light field). But when N is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the 1/N1/2 (1/N3/2) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the 1/N scaling for the linear phase and 1/N2 scaling for the nonlinear phase.
We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent k. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
We propose a method to generate the multi-mode entangled catalysis squeezed vacuum states (MECSVS) by embedding the cross-Kerr nonlinear medium into the Mach—Zehnder interferometer. This method realizes the exchange of quantum states between different modes based on Fredkin gate. In addition, we study the MECSVS as the probe state of multi-arm optical interferometer to realize multi-phase simultaneous estimation. The results show that the quantum Cramer—Rao bound (QCRB) of phase estimation can be improved by increasing the number of catalytic photons or decreasing the transmissivity of the optical beam splitter using for photon catalysis. In addition, we also show that even if there is photon loss, the QCRB of our photon catalysis scheme is lower than that of the ideal entangled squeezed vacuum states (ESVS), which shows that by performing the photon catalytic operation is more robust against photon loss than that without the catalytic operation. The results here can find applications in quantum metrology for multiparatmeter estimation.