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 $\sim 1/|\omega|^p$, with $p>1$. With no other assumptions, we show that the mean-square error has a lower bound scaling as $1/{\cal N}^{2(p-1)/(p+1)}$, where ${\cal N}$ 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.
Highlights
The probabilistic nature of quantum mechanics imposes fundamental limits to hypothesis testing and parameter estimation [1,2,3,4,5]
For the quantum parameter estimation problem, let ρX be the density operator that describes the state of a quantum probe as a function of the unknown parameters X, and let ÊY ðyÞ be the positive operator-valued measure (POVM)
We have considered phase estimation as an example of the application of the quantum Bell-Ziv-Zakai bounds
Summary
The probabilistic nature of quantum mechanics imposes fundamental limits to hypothesis testing and parameter estimation [1,2,3,4,5]. For the measurement of a single optical phase parameter, the ultimate quantum limit to the mean-square error scales as 1=n2 , where nis the average photon number of the field which undergoes that phase shift. For a waveform XðtÞ with stationary Gaussian prior statistics and a power-law spectrum (∝ 1=jωjp ; p > 1), we derive a lower bound on the mean-square error with a 1=N 2ðp−1Þ=ðpþ1Þ scaling, where N is the mean photon flux. This proof confirms that the scaling previously proposed in Ref. Dk 1⁄4 1⁄2Xk ðYÞ − Xk : E1⁄2Dðu⊤ εÞ 1⁄4 E1⁄2ðu⊤ εÞ2 1⁄4 u⊤ Σu; and let Y be a column vector of observations given by
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