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arXiv:2203.09541 · Physical Review A

Multiparameter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization

Wojciech Górecki, Rafał Demkowicz-Dobrzański

Published Jul 18, 2022• 19 citations
arXivHTMLPDF

Abstract

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 $\frac{1}{2}$, 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.

DOI
10.1103/physreva.106.012424

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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 NNN entangled spin 12\frac{1}{2}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.
Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN 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 NNN is strictly constrained, I show that the final measurement uncertainty is πππ 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.
Estimating multiple unknown parameters simultaneously is essential for practical quantum sensing. However, it faces a fundamental challenge: the optimal strategy for estimating one parameter is often incompatible with that for another, making it impossible to simultaneously achieve the ultimate precision limits for all parameters. Here we develop a general and efficient computational framework that jointly optimizes probe states, control operations, and measurements across different strategy families, including parallel, sequential, and those with indefinite causal order. Our approach provides exact semidefinite-program formulations for several precision bounds, including the Holevo, Nagaoka-Hayashi, and quantum Cramér-Rao bounds. We demonstrate the capabilities of the framework in multiparameter magnetometry and frequency estimation, identifying optimal protocols within each class and revealing a strict hierarchy among the achievable performances of different classes in the multiparameter regime. The framework also directly incorporates resource constraints, such as energy budgets, enabling systematic investigation of experimentally realistic sensing scenarios. Furthermore, we develop a finite-memory optimization method for sequential strategies with restricted ancillary-memory dimension. By decomposing the protocol into initial probe preparation and intermediate control operations, this method provides a practical route to designing resource-constrained sequential sensing schemes. Our work establishes a versatile computational tool for determining fundamental precision limits and designing optimal quantum-sensing protocols in complex multiparameter settings.
We explored the general characteristics of a Sagnac interferometer in a multiparameter estimation process. We find that in the two-parameter estimation scenario, one cannot make both parameter measurement results reach the Heisenberg limit (HL) simultaneously when the input resources are maximally entangled. Only one of the parameters' uncertainty can approach the HL while the other is only scaled by the standard quantum limit (SQL). We also discussed the constraint conditions that make the quantum Cramer-Rao bound saturable. These constraint conditions would prompt one to choose proper evolution time and optimal input state. Under the constraint conditions, we find that the HL result obtained in the two parameter scenario would catch up with or even be more precise than that acquired by the single parameter measurement process in some special cases. Such general features about the Sagnac system revealed in our work may have a reference value in actual experiments.
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.
Multiparameter quantum metrology is ubiquitous in physics, yet it remains fundamentally challenging. As reflected by the uncertainty principle, distinct unknown parameters generally cannot be estimated simultaneously at their individual ultimate precision limits. The quantum Cramér--Rao bound (QCRB) and the Holevo Cramér--Rao bound (HCRB) represent the ultimate precision bounds in multiparameter quantum metrology. Here we introduce new formulations of these bounds based on quantum state purification. We show that, for any mixed state, their values are directly related to those of its purification, provided that nuisance parameters are introduced on the environmental system. Leveraging this connection, we develop a remarkably simple method for asymptotically attaining either the HCRB or twice the QCRB for arbitrary mixed states, using random purification channels and individual measurements.