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arXiv:2609.12857

Optimal Hamiltonian Parameter Estimation in the Presence of Nuisance Parameters

Zhiyao Hu, Haidong Yuan, Liang Jiang, Zain H. Saleem

quant-phcs.ITSubmitted Sep 11, 2026• Announced Sep 14, 2026• 0 citations
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Abstract

In many sensing applications, the quantity of interest is not the only unknown, there are also additional unknown parameters, known as nuisance parameters, that affect the precision of estimation. While the ultimate local precision limit for a target parameter is well understood in the absence of nuisance parameters, the problem becomes significantly more challenging when they are present. In this work, we develop a framework for optimal Hamiltonian parameter estimation in the presence of nuisance parameters. We introduce an effective generator that captures the influence of nuisance parameters on the target precision, providing an explicit characterization of the ultimate precision limit for estimating the target parameter. Finally, we provide explicit optimal protocols, including probe state, control, and measurement that saturate this fundamental limit.

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21 pages, 1 figure

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In many sensing applications, the quantity of interest is not the only unknown, there are also additional unknown parameters, known as nuisance parameters, that affect the precision of estimation. While the ultimate local precision limit for a target parameter is well understood in the absence of nuisance parameters, the problem becomes significantly more challenging when they are present. In this work, we develop a framework for optimal Hamiltonian parameter estimation in the presence of nuisance parameters. We introduce an effective generator that captures the influence of nuisance parameters on the target precision, providing an explicit characterization of the ultimate precision limit for estimating the target parameter. Finally, we provide explicit optimal protocols, including probe state, control, and measurement that saturate this fundamental limit.
Estimation of physical parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the corresponding bound for estimating a function of multiple parameters-each coupled to distinct and possibly non-commuting generators-remains unknown in general. Here, we derive the ultimate quantum limit and present an estimation protocol for any function of parameters in a general Hamiltonian that attains this bound. We show that, although the task is fundamentally a multiparameter problem, our tight bound reduces to an optimized single-parameter quantum Cramér-Rao bound, even for arbitrary generator sets. Our result unifies and extends previous works, providing a general framework for optimal function estimation in quantum systems.
We establish the ultimate precision limits for estimating a function of multiple Hamiltonian parameters in the presence of Markovian noise. By reducing multiparameter function estimation to an optimization over effective single-parameter embeddings, we derive tight bounds on the quantum Fisher information. We identify a necessary and sufficient functional Hamiltonian-not-in-Lindblad-span condition for Heisenberg-limited scaling in time. When this condition holds, we construct a code that simultaneously removes the noise and nuisance Hamiltonian parameters while preserving the target signal. When it fails, we derive the optimal standard quantum-limit coefficient and show that it is asymptotically attainable using approximate quantum error correction. Finally, we demonstrate that direct function estimation can substantially outperform estimating all parameters individually and subsequently evaluating the function. This advantage can scale with both the evolution time and the number of parameters.
Abstract In parameter estimation, nuisance parameters refer to parameters that are not of interest but nevertheless affect the precision of estimating other parameters of interest. For instance, the strength of noises in a probe can be regarded as a nuisance parameter. Despite its long history in classical statistics, the nuisance parameter problem in quantum estimation remains largely unexplored. The goal of this article is to provide a systematic review of quantum estimation in the presence of nuisance parameters, and to supply those who work in quantum tomography and quantum metrology with tools to tackle relevant problems. After an introduction to the nuisance parameter and quantum estimation theory, we explicitly formulate the problem of quantum state estimation with nuisance parameters. We extend quantum Cramér–Rao bounds to the nuisance parameter case and provide a parameter orthogonalization tool to separate the nuisance parameters from the parameters of interest. In particular, we put more focus on the case of one-parameter estimation in the presence of nuisance parameters, as it is most frequently encountered in practice.
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.
Abstract In this paper, we analyze quantum-state estimation problems when some of the parameters are of no interest to be estimated. In classical statistics, these irrelevant parameters are called nuisance parameters, and this problem is of great importance in many practical applications of statistics. However, little is known regarding the effects of nuisance parameters in quantum-state estimation problems. The main contribution of this paper is first to formulate the nuisance parameter problem in the quantum-state estimation theory for the finite sample size case. We then derive the precision limit for the parameters of interest based on the concept of locally unbiasedness for the parameters of interest. We also propose a method of how to eliminate the nuisance parameters to obtain an upper bound for estimation error of the parameters of interest. Based on the proposed method, we clarify an intrinsic tradeoff relation to estimate the nuisance parameters and parameters of interest. The general qubit models are examined in detail to emphasize that we cannot ignore the effects of the nuisance parameters in general. Several examples are worked out to illustrate our findings.
A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.