Abstract We explore the general characteristics of a matter-wave Sagnac interferometer in a two-parameter estimation scheme. We find that the measurement precisions of both parameters cannot reach the Heisenberg limit (HL) simultaneously when the input state is maximally entangled. Only one of the parameters’ uncertainties can approach the HL while the other is scaled by the standard quantum limit. We provide the conditions with which the measurement precision of the specific parameter can reach the HL. We also discuss and figure out the concrete expressions of the constraint conditions for saturating the quantum Cramér–Rao bound. To satisfy these constraint conditions, the evolution time has to be a series of discrete values. Additionally, we calculate the variances of the parameters through some examples under these constraint conditions. The results provided in our work show some intrinsic features of the matter-wave Sagnac interferometer for the two-parameter estimation, which can be valuable in actual experiments.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of
N resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with
N better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources
N is strictly constrained, I show that the final measurement uncertainty is
π times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multiparameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cramér−Rao bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cramér−Rao bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.
Achieving the ultimate quantum precision in the estimation of multiple physical parameters simultaneously is a challenge in quantum metrology due to fundamental limitations and experimental challenges in harnessing the necessary quantum resources. We propose an experimentally feasible scheme to reach Heisenberg-scaling precision in the simultaneous estimation of two unknown phase parameters in a Mach-Zehnder interferometer by using squeezed and coherent states of light as input and homodyne detections at the outputs. Our results open a new research paradigm in harnessing the full quantum advantage of multiparameter estimation in distributed quantum sensing and quantum information processing, as enabled by multiphoton quantum interference with scalable experimental resources.
Quantum metrology achieves Heisenberg-limited precision in single-parameter estimation, but its multiparameter extension is fundamentally constrained by both parameter-encoding and measurement incompatibility. Noncommuting signal generators may cause incompatible parameter-encoding, preventing the quantum Fisher information matrix from simultaneously achieving the Heisenberg scale for all parameters. Due to incompatible optimal measurements, the classical Fisher information matrix represents the practical attainable precision. Here, we introduce a multiparameter metrology framework based on indefinite evolution (IE), in which different control operations and signal reversal are placed in a coherent superposition. For a single-qubit probe with mutually orthogonal signal generators, IE enables compatible parameter encoding and optimal measurement without the signal reversal. For parallel generators, where only signal reversal realized by its generator is available, IE can achieve the same performance. We further extend this mechanism to noisy, many-body, and high-dimensional probes, and establish general conditions for achieving the simultaneous Heisenberg-limit. In contrast, definite evolution cannot achieve the same performance under compatible optimal measurements, even when signal reversal is available. Our results identify IE as an operational resource for overcoming multiparameter incompatibility and open a route toward attainable Heisenberg-limited sensing in interferometric platforms.
We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms: one where the measurement is repeated many times (so the Cram\'er-Rao bound is guaranteed to be asymptotically saturable). and the second one where all the resources are allocated into one experimental realization (analyzed with the mimimax approach). We investigate the potential advantage of measuring all the parameters simultaneously compared to estimating them individually, while spending the same total amount of resources. We show that in general the existence of such an advantage, its magnitude, and conditions under which it occurs depends on which of the two paradigms has been chosen. In particular, for the problem of magnetic field sensing using
N entangled spin
21, we show that the predictions based purely on the Cram\'er-Rao formalism may be overly pessimistic in this matter: the minimax approach reveals the superiority of measuring all the parameters jointly whereas the Cram\'er-Rao approach indicates lack of such an advantage.