Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation

Abstract

We propose quantum versions of the Bell-Ziv-Zakai lower bounds on the error in multiparameter estimation. As an application we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power law spectrum ∼ 1/|ω| p , with p > 1. With no other assumptions, we show that the mean-square error has a lower bound scaling as 1/N 2(p−1)/(p+1) , where N is the time-averaged mean photon flux. Moreover, we show that this accuracy is achievable by sampling and interpolation, for any p > 1. This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.

Subdiffraction incoherent optical imaging via spatial-mode demultiplexing

“…The quantum optimality of SPADE for general imaging is in particular an interesting question. These daunting problems may be attacked by more advanced methods in quantum metrology [43][44][45][46][75][76][77][78], quantum state tomography [79][80][81], compressed sensing [55][56][57]81], and photonics design [52][53][54]. For the TEM scheme, this implies that each qth channel contains information about not only q q 2 but also the higher-order moments.…”

Quantum parameter estimation with general dynamics

“…; [1][2][3][4][5][6][7][8][9][10][11] this has gained increasing attention in recent years. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] A typical situation in quantum parameter estimation is to estimate the value of a continuous parameter x encoded in some quantum state ρ x of the system. To estimate the value, one needs to first perform measurements on the system, which, in the general form, are described by Positive Operator Valued Measurements (POVM), {E y }, which provides a distribution for the measurement results p(y|x) = Tr(E y ρ x ).…”

Multiparameter estimation along quantum trajectories with sequential Monte Carlo methods

“…A full description of the problem involves starting with a prior probability density for the parameters one wishes to determine and then using Bayes' theorem to continually update this probability density from the stream of measurement results as they are obtained. A number of authors have considered this problem [15,[18][19][20][21][22][23][24][25][26][27][28]. This is of particular interest when the parameters of a system change slowly with time, and one wishes to be able to track the variations in the parameters.…”

“…The first papers on the subject of Hamiltonian parameter estimation via continuous measurements were concerned mainly with deriving the Hybrid SME and applying it to the estimation of a single parameter [18,19]. Subsequently, Tsang and collaborators considered the more general problem of smoothing in which a time-varying parameter (a signal or wave-form) is estimated from all the measurement results obtained, and determined the ultimate limits to this procedure [21][22][23][24]26]. An alternative and interesting approach to the problem was proposed recently by Bassa et al [27].…”