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

We propose quantum versions of the Bell-Ziv-Zakai lower bounds for 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 $1/{\mathcal{N}}^{2(p-1)/(p+1)}$, where $\mathcal{N}$ is the time-averaged mean photon flux. Moreover, we show that this scaling 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.

Popular Summary

Quantum mechanics puts fundamental limits (e.g., Heisenberg’s uncertainty principle) on how well scientists can estimate unknown quantities with finite resources. An important task for optical and atomic sensing applications is to estimate a signal that fluctuates in time, but the proof of its quantum limit due to finite energy has, up until now, eluded researchers. We have proven such a limit for a time-varying, stochastic optical phase signal, and we have also shown how to approach it.

The majority of previous studies have considered quantum bounds on the estimation of a single, fixed parameter. However, some critical scientific goals, such as gravitational-wave astronomy, require the estimation of signals that change unpredictably in time. This situation is far more difficult to analyze because it involves an infinite number of unknown but correlated parameters. For this purpose, we develop a new tool—a quantum generalization of the Bell-Ziv-Zakai bound—to prove the quantum limit on the error in the estimation of an optical phase. Moreover, we develop a scheme that is able to follow the limit closely, thus showing that the limit is not only rigorous but also relevant to reality. We apply the quantum Bell-Ziv-Zakai bound to the estimation of a time-varying optical phase, but this powerful technique could also be applied to many other measurement tasks.

We expect that our methodology can be used for determining ultimate quantum limits to optical imaging, time-dependent magnetometry, and optomechanical sensing.

Figure 1

The erfc function and a lower bound using the triangle function $\mathrm{\Lambda }$.

Figure 2

The quantum estimation problem for commuting generators. The initial state $\rho$ may be entangled, and each subsystem or mode undergoes a displacement $X_j$ generated by operator ${\widehat{g}}_j$, with $[{\widehat{g}}_j,{\widehat{g}}_k]=0$. The output may be measured via some general joint measurement ${\widehat{E}}_{\mathit{Y}}(\mathit{y})$.

Figure 3

A lower bound on the cosine function.

Figure 4

The continuous waveform estimation problem. The signal $X(t)$ varies continuously, and the system may be approximated by regarding the state as consisting of (typically entangled) modes of length $\delta t$, over each of which $X(t)$ does not vary significantly.

Figure 5

A convex function that appears in Eq. (4.37).

Figure 6

A pulsed phase measurement scheme to achieve the optimal scaling.

Figure 7

Top panel: The modulo function $[z{]}_{2\pi }$. Bottom panel: Square of the modulo function.