Uncertainty and trade-offs in quantum multiparameter estimation

Abstract

Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In Quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér-Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state independent uncertainty relation between the parameters of a qubit system.Ever since its first formulation, the uncertainty principle has seen many refinements and clarifications. As quantum theory developed, its state-of-the-art concepts and mathematical tools were used to formulate in precise terms the ideas which were put forward in Heisenberg's 1927 paper [1]. As a result, our current understanding of the uncertainties inherent in quantum mechanics is spelled out in a collection of theorems pertaining to well defined operational tasks.Soon after Heisenberg's paper, rigorous proofs of his uncertainty relations were formulated [2][3][4]. Those are usually referred to as preparation uncertainty relations. Most well known is the relation due to Weyl and Robertson arXiv:2002.05961v1 [quant-ph]

On the Quantumness of Multiparameter Estimation Problems for Qubit Systems

“…We believe that our work together with other complementary approaches, such as the one pursued in [ 64 ] where trade-off surfaces are derived via the SLD- and the Holevo-bound, will help in shedding new light on the relationship between quantum uncertainty relations, incompatibility and multi-parameter quantum metrology.…”

Quantum Cramér-Rao bound based on the Bures distance

“…Helstrom derived the quantum Cramér-Rao bound (QCRB) by defining the quantum Fisher information (QFI) matrix as an analogue of the classical Fisher information matrix [5][6][7][8][9]. Due to the measurement incompatibility caused by Heisenberg's uncertainty principle [10], the quantum estimation problems for multiple parameters are much more intricate than the classical estimation problems [11][12][13][14][15][16][17][18][19]. For the estimations of a single parameter, the QCRB based on the QFI gives a rather satisfactory approach to revealing the ultimate quantum limit of estimation precision.…”

Efficient computation of the Nagaoka—Hayashi bound for multi-parameter estimation with separable measurements

“…Many of the applications of quantum estimation require the simultaneous measurement of multiple parameters 19,20 , which in general will not commute with each other. This means that a measurement that is optimal for one parameter may not be optimal for another which limits the precision with which we can measure them simultaneously [21][22][23][24][25] . Thus, in an effort to fully exploit quantum resources in real-world applications, there has been great experimental [26][27][28][29][30] and theoretical interest in quantum multiparameter estimation [31][32][33][34][35][36][37][38][39][40][41][42][43] .…”