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arXiv:2310.06767 · Journal of Physics A Mathematical and Theoretical

Optimal estimation of pure states with displaced-null measurements

Federico Girotti, Alfred Godley, Mădălin Guţǎ

Published May 15, 2024• 6 citations
arXivHTMLPDF

Abstract

Abstract We revisit the problem of estimating an unknown parameter of a pure quantum state, and investigate ‘null-measurement’ strategies in which the experimenter aims to measure in a basis that contains a vector close to the true system state. Such strategies are known to approach the quantum Fisher information for models where the quantum Cramér-Rao bound (QCRB) is achievable but a detailed adaptive strategy for achieving the bound in the multi-copy setting has been lacking. We first show that the following naive null-measurement implementation fails to attain even the standard estimation scaling: estimate the parameter on a small sub-sample, and apply the null-measurement corresponding to the estimated value on the rest of the systems. This is due to non-identifiability issues specific to null-measurements, which arise when the true and reference parameters are close to each other. To avoid this, we propose the alternative displaced-null measurement strategy in which the reference parameter is altered by a small amount which is sufficient to ensure parameter identifiability. We use this strategy to devise asymptotically optimal measurements for models where the QCRB is achievable. More generally, we extend the method to arbitrary multi-parameter models and prove the asymptotic achievability of the the Holevo bound. An important tool in our analysis is the theory of quantum local asymptotic normality which provides a clear intuition about the design of the proposed estimators, and shows that they have asymptotically normal distributions.

DOI
10.1088/1751-8121/ad4c2b

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Abstract We revisit the problem of estimating an unknown parameter of a pure quantum state, and investigate ‘null-measurement’ strategies in which the experimenter aims to measure in a basis that contains a vector close to the true system state. Such strategies are known to approach the quantum Fisher information for models where the quantum Cramér-Rao bound (QCRB) is achievable but a detailed adaptive strategy for achieving the bound in the multi-copy setting has been lacking. We first show that the following naive null-measurement implementation fails to attain even the standard estimation scaling: estimate the parameter on a small sub-sample, and apply the null-measurement corresponding to the estimated value on the rest of the systems. This is due to non-identifiability issues specific to null-measurements, which arise when the true and reference parameters are close to each other. To avoid this, we propose the alternative displaced-null measurement strategy in which the reference parameter is altered by a small amount which is sufficient to ensure parameter identifiability. We use this strategy to devise asymptotically optimal measurements for models where the QCRB is achievable. More generally, we extend the method to arbitrary multi-parameter models and prove the asymptotic achievability of the the Holevo bound. An important tool in our analysis is the theory of quantum local asymptotic normality which provides a clear intuition about the design of the proposed estimators, and shows that they have asymptotically normal distributions.
We consider estimation of a single unknown parameter embedded in a quantum state. Quantum Cramér-Rao bound (QCRB) is the ultimate limit of the mean squared error for any unbiased estimator. While it can be achieved asymptotically for a large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. Prior work addresses this paradox using a two-stage approach: in the first stage, a preliminary estimate is obtained by applying, on a vanishing fraction of quantum state copies, a sub-optimal measurement that does not depend on the parameter of interest. In the second stage, the preliminary estimate is used to construct the QCRB-achieving measurement that is applied to the remaining quantum state copies. This is akin to two-step estimators for classical problems with nuisance parameters. Unfortunately, the original analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators for single-parameter problems at the cost of slightly weakening the asymptotic properties of the two-stage method. We also account for nuisance parameters. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.
In quantum computation, amplitude estimation is a fundamental subroutine that is utilized in various quantum algorithms. A general important task of such estimation problems is to characterize the estimation lower bound, which is referred to as quantum Cram\'er-Rao bound (QCRB), and to construct an optimal estimator that achieves QCRB. This paper studies the amplitude estimation in the presence of depolarizing noise with unknown intensity. The main difficulty in this problem is that the optimal measurement depends on both the unknown quantum state and the amplitude we aim to estimate. To deal with these issues, we utilize the variational quantum circuits to approximate the (unknown) optimal measurement basis combined with the two-step adaptive estimation strategy which was proposed in the quantum estimation theory. We numerically show that the proposed method can nearly attain the QCRB.
For a given quantum state used in sensing, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the precision achievable by an unbiased estimator of an unknown parameter, determined by the inverse of the quantum Fisher information (QFI). The QFI serves as an upper bound on the classical Fisher information (CFI), representing the maximum extractable information about the unknown parameter from measurements on a physical system. Thus, a central goal in quantum parameter estimation is to find a measurement, described by a POVM, that saturates the QFI (achieves maximum CFI), and thereby achieves the QCRB. In the idealization that one uses pure states and unitary encodings for sensing, discontinuities can appear in the CFI but not the QFI. In this article, we demonstrate that these discontinuities are important features, quantifying how much Fisher information is lost in the presence of noise. We refer to this as the Fisher information "fragility". We present a simple framework for understanding how discontinuities increase fragility through Jensen's inequality, and demonstrate how one can use this framework to design more robust POVMs for quantum advantage in metrology.
Quantum measurement plays a crucial role in quantum metrology. Due to the limitations of experimental capabilities, collectively measuring multiple copies of probing systems can present significant challenges. Therefore, the concept of locality in quantum measurements must be considered. In this work, we investigate the possibility of achieving the quantum Cramér-Rao bound (QCRB) through local measurements (LMs). We first demonstrate that if there exists a LM to saturate the QCRB for qubit systems, then we can construct another rank-1 local projective measurement to saturate the QCRB. In this sense, rank-1 local projective measurements are sufficient to analyze the problem of saturating the QCRB. For pure qubits, we propose two necessary and sufficient methods to determine whether and how a given parameter estimation model can achieve the QCRB through LMs. The first method, dubbed the iterative matrix partition method and based on unitary transformations that render the diagonal entries of a traceless matrix vanish, elucidates the underlying mathematical structure of LMs as well as the local measurements with classical communications (LMCC), generalizing the result by Zhou . [], which only holds for the later case. We clarify that the saturation of the QCRB through LMs for the Greenberger–Horne–Zeilinger-encoded states is actually due to the self-similar structure in this approach. The second method, dubbed the hierarchy of orthogonality conditions and based on the parametrization of rank-1 measurements for qubit systems, allows us to construct several examples of saturating QCRBs, including the three-qubit W states and the N -qubit W states ( N ≥ 3 ). Our findings offer insights into achieving optimal performance in quantum metrology when measurement resources are limited.
Abstract This review aims at gathering the most relevant quantum multi-parameter estimation methods that go beyond the direct use of the quantum Fisher information concept. We discuss in detail the Holevo Cramér–Rao bound, the quantum local asymptotic normality approach as well as Bayesian methods. Even though the fundamental concepts in the field have been laid out more than forty years ago, a number of important results have appeared much more recently. Moreover, the field drew increased attention recently thanks to advances in practical quantum metrology proposals and implementations that often involve estimation of multiple parameters simultaneously. Since the topics covered in these review are spread in the literature and often served in a very formal mathematical language, one of the main goals of this review is to provide a largely self-contained work that allows the reader to follow most of the derivations and get an intuitive understanding of the interrelations between different concepts using a set of simple yet representative examples involving qubit and Gaussian shift models.
Determining when the multiparameter quantum Cramér--Rao bound (QCRB) is saturable with experimentally relevant single-copy measurements is a central open problem in quantum metrology. Here we establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model, i.e., the existence of complete (generally nonorthogonal) bases in which all corresponding diagonal matrix elements vanish. This formulation yields a geometric characterization: optimal rank-one measurement vectors are confined to a subspace orthogonal to a state-determined Hermitian span. This provides a direct criterion to construct optimal Positive Operator-Valued Measures(POVMs). We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104(2019)] becomes necessary and sufficient for the saturation of the QCRB, demonstrate that this condition is not always sufficient, and prove the counter-intuitive uselessness of informationally-complete POVMs.