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arXiv:1806.07337 · Physical Review A

Optimal measurements for quantum multiparameter estimation with general states

Jing Yang, Shengshi Pang, Yiyu Zhou, Andrew N. Jordan

Published Sep 3, 2019• 86 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.100.032104

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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.
We formulate multiparameter quantum estimation in the parametric and semiparametric setting. While the Holevo Cramér-Rao bound (CRB) requires no substantial modifications in moving from the former to the latter, we generalize the Helstrom CRB appropriately. We show that the Holevo CRB cannot be greater than twice the generalized Helstrom CRB. We also present a tighter, intermediate, bound. Finally, we show that for parameters encoded in the first moments of a Gaussian state there always exists a Gaussian measurement that gives a classical Fisher information matrix that is one-half of the quantum Fisher information matrix.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.
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.
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.
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.
The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general ppp-local measurements, which are the measurements that can be performed collectively on at most ppp copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cram\'er-Rao bound under ppp-local measurements, which recovers the partial commutative condition at p=1p=1p=1 and the weak commutative condition at $p=\ensuremath{\infty}$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.