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.
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.
Practical Sensitivity Bound for Multiple Phase Estimation with Multi‐Mode N00NN00N States
Abstract Quantum enhanced multiple phase estimation is essential for various applications in quantum sensors and imaging. For multiple phase estimation, the sensitivity enhancement is dependent on both quantum probe states and measurement. It is known that multi‐mode states can outperform other probe states for estimating multiple phases. However, it is generally not feasible in practice to implement an optimal measurement to achieve the quantum Cramer–Rao bound (QCRB) under a practical measurement scheme using a multi‐mode beam splitter in interferometric phase estimation. Here, a strategy to achieve the best practical sensitivity by optimizing both mode‐amplitudes of multi‐mode states and a split ratio of a multi‐mode beam splitter is investigated. Then, it is experimentally demonstrated that the best sensitivity is achieved when an amplitude‐balanced multi‐mode state and a multi‐mode beam splitter with an unbalanced ratio are used in three‐mode interferometric phase estimation. The results show that the lower QCRB cannot guarantee better sensitivity under a practical measurement scheme, thus it is more desirable to enhance the practical sensitivity rather than the QCRB. It is believed that this strategy can provide a powerful tool for practical applications in multiple phase estimation.
Quantum metrology can achieve enhanced sensitivity for estimating unknown parameters beyond the standard quantum limit. Recently, multiple-phase estimation exploiting quantum resources has attracted intensive interest for its applications in quantum imaging and sensor networks. For multiple-phase estimation, the amount of enhanced sensitivity is dependent on quantum probe states, and multi-mode N00N states are known to be a key resource for this. However, its experimental demonstration has been missing so far since generating such states is highly challenging. Here, we report generation of multi-mode N00N states and experimental demonstration of quantum enhanced multiple-phase estimation using the multi-mode N00N states. In particular, we show that the quantum Cramer-Rao bound can be saturated using our two-photon four-mode N00N state and measurement scheme using a 4 × 4 multi-mode beam splitter. Our multiple-phase estimation strategy provides a faithful platform to investigate multiple parameter estimation scenarios.
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.
The minimum possible uncertainty when estimating a parameter, in this case transmission, is bounded by the quantum Cramér-Rao bound (QCRB). This bound is independent of the measurement technique and as result can be used to quantify how close a measurement is to the optimal one for a given system and probing state, as an optimal measurement saturates the QCRB. The ability to perform transmission measurements through an optical system at the QCRB can lead to enhancements in the calibration of optimal states for interferometers, the characterization of high efficiency photodetectors, or the sensitivity of sensing devices based on transmission, such as plasmonic sensors or spectroscopy. We show theoretically that the ultimate sensitivity limit per photon in the estimation of transmission, achievable with Fock states or vacuum two-mode squeezed states (vTMSS), can be approached through the use of bright quantum states of light, such as bright single-mode squeezed states (bSMSS) or bright two-mode squeezed states (bTMSS). The use of bright quantum states has the advantage that they can be generated with a significantly larger number of photons, which leads to an absolute sensitivity that can be many orders of magnitude larger than the one achieved with Fock states or vTMSS. We identify a measurement strategy that saturates the QCRB for these bright quantum states and shown experimentally that transmission estimation at the QCRB can be achieved with bTMSS through optimized intensity difference measurements.
How to find the specific measurement scheme to approach the quantum Cramér-Rao bound (QCRB) is an important issue in quantum metrology, especially for the continuous-variable optical interferometers input with non-classical states. In the ideal situation (i.e., without any photon loss), this can be usually achieved by using the parity measurement, which, however, does not work well when the photon loss occur. Alternatively, in this work we find that the coincidence measurements are more robust to the photon loss than the parity measurement. Importantly, in certain cases, the sensitivity with the coincidence measurement can even approach very closely to the QCRB despite of the photon loss. Though the results are obtained based on the analysis with the typical two-photon twin-Fock state, it might be also generalized for the other input ones. Thus, our work may provide an effective measurement scheme for beating the shot-noise limit in practical experiments.