Quantum Cramer-Rao Bound Research Articles

Cyclic nonlinear interferometry with entangled non-Gaussian spin states

We propose an efficient nonlinear readout scheme for entangled non-Gaussian spin states based on the intrinsic quasicyclic dynamics of interacting spin-$\frac{1}{2}$ systems. We focus on two well-known spin models of twist-and-turn (TNT) and two-axis-countertwisting (TACT), where the entangled non-Gaussian spin state can be generated by spin dynamics starting from unstable fixed points. In the TNT model, the non-Gaussian probe state evolves directly back to the vicinity of the initial state during the subsequent time-forward evolution for path recombining, accompanied by quantum magnification of an encoded signal and refocusing of the associated quantum noise. Based on low-order moment measurement, we find that the optimal metrological gain nearly saturates the quantum Cram\'er-Rao bound (QCRB) and follows the Heisenberg scaling. For the TACT case, the QCRB can also be nearly approached when the state converges to either of the two unstable fixed points corresponding to the initial state or its orthogonal coherent state, respectively. The latter case goes beyond previous studies where tracing back to or crossing the initial states was mostly considered. The present protocol does not require time reversal as in typical nonlinear interferometries and it also avoids complicated measurement of nonlinear observables or full probability distributions. The operational approach we discuss presents a practical way to realize high-precision and detection-noise-robust quantum metrology with entangled non-Gaussian spin states.

Tilt Measurement at the Quantum Cramer–Rao Bound Using a Higher-Order Hermite–Gaussian Mode

The quantum Cramer–Rao bound (QCRB) provides an ultimate precision limit in parameter estimation. The sensitivity of spatial measurements can be improved by using the higher-order Hermite–Gaussian mode. However, to date, the QCRB-saturating tilt measurement has not been realized. Here, we experimentally demonstrate tilt measurements using a higher-order HG40 mode as the probe beam. Using the balanced homodyne detection with an optimal local beam, which involves the superposition of high-order HG30 and HG50 modes, we demonstrate the precision of the tilt measurement approaching the QCRB. The signal-to-noise ratio of the tilt measurement is enhanced by 9.2 dB compared with the traditional method using HG00 as the probe beam. This scheme is more practical and robust to losses, which has potential applications in areas such as laser interferometer gravitational-wave observatories and high-sensitivity atomic force microscopes.

Optimal strategy for multiple-phase estimation under practical measurement with multimode NOON states

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 metrology with multimode Gaussian states of multiple point sources

A unified theoretical framework is developed for measuring multimode Gaussian states of multiple point sources. The spatial and temporal mode functions are found analytically by aid of the circular symmetry of the object plane and aperture, as well as by the rectangular spectral density function of the background. They relate the modal operators and statistical moments to the locations of the point sources through the prolate spheroidal functions. An experimentally straightforward implementation is proposed, which consists of a nonabsorptive polarization- and phase-modulating spatial light modulator (SLM), three lenses, and a polarization beam splitter. Interestingly, it is found that the spatial modal annihilation operators are related to the modal field operators in the image plane by a simple complex-valued scale factor. The quantum Fisher information matrix and quantum Cram\'er-Rao bound (QCRB) are also discussed. The numerical calculation of QCRB is completed for the case where the spatially multimode aperture fields are prepared by the spatial propagation from temporally single-mode thermal states, coherent states, or quadrature squeezed states of the point sources in the object plane. The results show that the Rayleigh limit can always be surpassed by means of increasing the mean photon rate or the squeeze parameter.

Practical Sensitivity Bound for Multiple Phase Estimation with Multi‐Mode N00N$N00N$ States

AbstractQuantum 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.

Enhancing phase sensitivity with number state filtered coherent states

Number state filtered coherent states are a class of nonclassical states obtained by removing one or more number states from a coherent state. Phase sensitivity of an interferometer is enhanced if these nonclassical states are used as input states. The optimal phase sensitivity, which is related to the quantum Cramer-Rao bound (QCRB) for the input state, improves beyond the standard quantum limit. It is argued that removal of more than one suitable number state leads to better phase sensitivity. As an important limiting case in this context, the even and odd coherent states, where the odd and even number states are filtered from coherent state respectively, are considered. The optimal phase sensitivity for these limiting cases equals that of the squeezed vacuum. It is observed that the improvement in phase sensitivity is not in direct proportion to the nonclassicality of the input states.

Quantum multiparameter estimation with generalized balanced multimode NOON-like states

The simultaneous multi-parameter estimation problem using a class of multi-mode 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 multi-mode NOON state. The analytical form for the quantum Cramer-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. We also illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios. Finally, we give an experimental setup for producing a two-mode entangled state that can beat the NOON state in quantum metrology.

Quantum phase estimation using path-symmetric entangled states.

We study the sensitivity of phase estimation using a generic class of path-symmetric entangled states |φ〉|0〉 + |0〉|φ〉, where an arbitrary state |φ〉 occupies one of two modes in quantum superposition. With this generalization, we identify the fundamental limit of phase estimation under energy constraint that is characterized by the photon statistics of the component state |φ〉. We show that quantum Cramer-Rao bound (QCRB) can be indefinitely lowered with super-Poissonianity of the state |φ〉. For possible measurement schemes, we demonstrate that a full photon-counting employing the path-symmetric entangled states achieves the QCRB over the entire range [0, 2π] of unknown phase shift ϕ whereas a parity measurement does so in a certain confined range of ϕ. By introducing a component state of the form , we particularly show that an arbitrarily small QCRB can be achieved even with a finite energy in an ideal situation. This component state also provides the most robust resource against photon loss among considered entangled states over the range of the average input energy Nav > 1. Finally we propose experimental schemes to generate these path-symmetric entangled states for phase estimation.

Fundamental Quantum Limit to Waveform Estimation

We derive a quantum Cramér-Rao bound (QCRB) on the error of estimating a time-changing signal. The QCRB provides a fundamental limit to the performance of general quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers. We apply the QCRB to the problem of force estimation via continuous monitoring of the position of a harmonic oscillator, in which case the QCRB takes the form of a spectral uncertainty principle. The bound on the force-estimation error can be achieved by implementing quantum noise cancellation in the experimental setup and applying smoothing to the observations.