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Physical Review A

Fisher information of sequential measurements and the optimization in composite systems

Jianning Li, Dianzhen Cui

Published Jul 17, 2023• 3 citations
Publisher

Abstract

In parameter estimation, the saturation of the quantum Cram\'er-Rao bound (QCRB) requests one to perform projective measurements on the eigenvectors of the symmetric logarithmic derivative (SLD). However, it maybe infeasible to saturate the QCRB in composite systems on account of the technological difficulties. In practice, it is necessary to perform sequential measurements on different subsystems to obtain a precise estimation. In this paper, we derive the expression of Fisher information (FI) for a sequential measurements scheme in composite systems with $N$ interacting subsystems. The result shows that the more subsystems used, the higher FI obtained due to the nonnegative property of FI. We illustrate the optimal measurements for different conditional density matrices in every subsystems are nonidentical, which depends on the previous outcomes. In addition, utilizing the convexity property of the quantum FI, we prove that the optimal projective measurements on the eigenvectors of the SLD of the subsystem reduced density matrix can be regarded as a trade off of the nonidentical optimal measurements of the conditional density matrices in every subsystems. In addition, importantly, we show the measurement sequence of the subsystem should be considered in the optimization to obtain a precise estimation for the sequential measurements scheme. Finally, we apply the theory to a nuclear spins control model where we concern the estimation of nuclear Larmor frequency. We should point out that the precision obtained by the sequential measurements scheme is no better than the theoretical result in the global system limited by the QCRB but more realistic, and better than the local measurements in one of the subsystems.

DOI
10.1103/physreva.108.012419

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In parameter estimation, the saturation of the quantum Cram'er-Rao bound (QCRB) requests one to perform projective measurements on the eigenvectors of the symmetric logarithmic derivative (SLD). However, it maybe infeasible to saturate the QCRB in composite systems on account of the technological difficulties. In practice, it is necessary to perform sequential measurements on different subsystems to obtain a precise estimation. In this paper, we derive the expression of Fisher information (FI) for a sequential measurements scheme in composite systems with NNN interacting subsystems. The result shows that the more subsystems used, the higher FI obtained due to the nonnegative property of FI. We illustrate the optimal measurements for different conditional density matrices in every subsystems are nonidentical, which depends on the previous outcomes. In addition, utilizing the convexity property of the quantum FI, we prove that the optimal projective measurements on the eigenvectors of the SLD of the subsystem reduced density matrix can be regarded as a trade off of the nonidentical optimal measurements of the conditional density matrices in every subsystems. In addition, importantly, we show the measurement sequence of the subsystem should be considered in the optimization to obtain a precise estimation for the sequential measurements scheme. Finally, we apply the theory to a nuclear spins control model where we concern the estimation of nuclear Larmor frequency. We should point out that the precision obtained by the sequential measurements scheme is no better than the theoretical result in the global system limited by the QCRB but more realistic, and better than the local measurements in one of the subsystems.
The attainability of the quantum Cram\'er-Rao bound (QCR), the ultimate limit in the precision of the estimation of a physical parameter, requires the saturation of the quantum information bound (QIB). This occurs when the Fisher information associated to a given measurement on the quantum state of a system which encodes the information about the parameter coincides with the quantum Fisher information associated to that quantum state. Braunstein and Caves [Phys. Rev. Lett. 72, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called the symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated, therefore requiring previous knowledge of the quantity one is trying to estimate. For this reason, it is important to investigate under which situation it is possible to saturate the QCR without previous information about the parameter to be estimated. Here, we show the complete solution to the problem of which are the initial pure states and which projective measurements allow the global saturation of the QIB, without the knowledge of the true value of the parameter, when the information about the parameter is encoded in the system by a unitary process.
Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When there is more than one parameter the ultimate precision in the mean square error given by the quantum Cramér-Rao bound is not necessarily achievable. For non-full rank quantum states, it was not known when this bound can be saturated (achieved) when only a single copy of the quantum state encoding the unknown parameters is available. This single-copy scenario is important because of its experimental/practical tractability. Recently, necessary and sufficient conditions for saturability of the quantum Cramér-Rao bound in the multiparameter single-copy scenario have been established in terms of i) the commutativity of a set of projected symmetric logarithmic derivatives and ii) the existence of a unitary solution to a system of coupled nonlinear partial differential equations. New sufficient conditions were also obtained that only depend on properties of the symmetric logarithmic derivatives. In this paper, key structural properties of optimal measurements that saturate the quantum Cramér-Rao bound are illuminated. These properties are exploited to i) show that the sufficient conditions are in fact necessary and sufficient for an optimal measurement to be projective, ii) give an alternative proof of previously established necessary conditions, and iii) describe general POVMs, not necessarily projective, that saturate the multiparameter QCRB. Examples are given where a unitary solution to the system of nonlinear partial differential equations can be explicitly calculated when the required conditions are fulfilled.
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.
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.
The quantum Cramér-Rao bound (QCRB) as the ultimate lower bound for precision in quantum parameter estimation is only known to be saturable in the multiparameter setting in special cases and under conditions such as full or average commutavity of the symmetric logarithmic derivatives (SLDs) associated with the parameters. Moreover, for general mixed states, collective measurements over infinitely many identical copies of the quantum state are generally required to attain the QCRB. In the important and experimentally relevant single-copy scenario, a necessary condition for saturating the QCRB in the multiparameter setting for general mixed states is the so-called partial commutativity condition on the SLDs. However, it is not known if this condition is also sufficient. This paper establishes necessary and sufficient conditions for saturability of the multiparameter QCRB in the single-copy setting in terms of the commutativity of a set of projected SLDs and the existence of a unitary solution to a system of nonlinear partial differential equations. New necessary conditions that imply partial commutativity are also obtained, which together with another condition become sufficient. Moreover, when the sufficient conditions are satisfied an optimal measurement saturating the QCRB can be chosen to be projective and explicitly characterized. An example is developed to illustrate the case of a multiparameter quantum state where the conditions derived herein are satisfied and can be explicitly verified.
The quantum Cram\'er-Rao theorem states that the quantum Fisher information bounds the best achievable precision in the estimation of a quantum parameter $\ensuremath{\xi}$. This is true, however, under the assumption that the measurement employed to extract information on $\ensuremath{\xi}$ is regular, i.e., neither its sample space nor its positive-operator valued elements depend on the (true) value of the parameter. A better performance may be achieved by relaxing this assumption. In the case of a general Hamiltonian parameter, i.e., when the parameter enters the system's Hamiltonian in a nonlinear way (making the energy eigenstates and eigenvalues $\ensuremath{\xi}$ dependent), a family of nonregular measurements, referred to as controlled energy measurements, is naturally available. We perform an analytic optimization of their performance, which enables us to compare the optimal controlled energy measurement with the optimal Braunstein-Caves measurement based on the symmetric logarithmic derivative. As the former may outperform the latter, the ultimate quantum bounds for general Hamiltonian parameters are different than those for phase (shift) parameters. We also discuss in detail a realistic implementation of controlled energy measurements based on the quantum phase estimation algorithm and work out a variety of examples to illustrate our results.