Quantum metrology: Background

Measurement is at the heart of science. The action of performing a measurement involves extracting information about a physical quantity. Every measurement results in an estimate of the measured quantity, and an uncertainty (or error) within which this estimate represents the true value of the quantity. For example, measuring the length of a pencil with a ruler provides an estimate for the true length of the pencil, and the uncertainty is determined by the resolution of the ruler.

In order to reduce the uncertainty of a measurement result, we can repeat the measurement $N$ times to get an average. In particular, if each individual measurement has an uncertainty $\sigma$ (and if the individual measurements are independent and their potential outcomes are identically distributed), then the uncertainty $\sigma_N$ associated with the average of the $N$ measurements is given by \[\sigma_N = \frac{\sigma}{\sqrt{N}},\] i.e., the uncertainty scales as $1/\sqrt{N}$. This is called the standard quantum limit, and corresponds to the smallest uncertainty attainable if only classical resources are used. As an example, if we wanted to reduce the uncertainty by a factor of 10, we would need to repeat the measurement 100 times.

The standard quantum limit is not the fundamental limit to measurement precision. The ultimate limit is set by measurement uncertainty in quantum mechanics, and is called the Heisenberg limit [1]. At the Heisenberg limit, the uncertainty scales with the number of times a system is probed such that \[\sigma_N \propto \frac{1}{N}.\] In this case, reducing the uncertainty by a factor of 10 would require the unknown quantity to be sampled only 10 times. This illustrates that the Heisenberg limit can provide a significant reduction in resources required to achieve a particular level of precision compared to the standard quantum limit, and hence is of fundamental importance in the context of measurement. Reaching Heisenberg-limited precision is one of the goals of quantum metrology, and can only be achieved with nonclassical strategies [2,3]. In particular, quantum effects (e.g., entanglement) are essential at the probe preparation stage in order to reach the fundamental limit to measurement precision.

Quantum metrology protocols are often formulated as a phase estimation problem, where the unknown quantity to be estimated is a phase shift $\varphi$ in an interferometer, and the probe consists of photons [4] (see Figure 1 for a simple example). In this context, the standard quantum limit is referred to as the shot-noise limit, and arises from the Poissonian statistics of classical light [1] (random fluctuations in the photon number). The shot-noise limit can be overcome by using nonclassical states of light, such as NOON states [5,6] or squeezed states [7,8,9], the latter having been used to enhance the sensitivity in gravitational-wave detection [10,11]. In addition, adaptive measurement strategies can be adopted, where feedback is used to gradually improve the estimate of the unknown phase. This includes online schemes, where the optimal feedback action is determined in real time based on results from a previous measurement [12,13] (e.g., Bayesian estimation), as well as offline schemes, where optimization algorithms are used to determine the optimal feedback sequence in advance of the measurement process [14,15]. Adaptive phase estimation has successfully been used to achieve Heisenberg-limited error scaling with single-photon states [16], where multiple applications of the phase shift have been used on the single photons to reach the precision limit without needing more complex entangled input states like NOON or squeezed states.

Figure 1: Diagram of a Mach-Zehnder interferometer consisting of two beamsplitters (BS), two mirrors (M), and an unknown phase shift $\varphi$ in one of the arms of the interferometer. The distribution of photon counts at detectors D1 and D2 depends on $\varphi$, which means that information about the phase can be inferred from the measured signal.

The new approach for achieving Heisenberg-limited precision

In Ref. [17], Simone Roncallo, Xi Lu, and Lorenzo Maccone propose a new scheme for reaching Heisenberg-limited precision in phase estimation. In their interferometric scheme, the authors use additional phases $\theta_i$ to control the probability distribution of the output photons, and choose $\theta_i$ such that the distribution approximates a square wave. This allows the phase estimation problem to be formulated as a binary search task consisting of multiple iterations. In iteration $j \in \{1,2,3,\ldots\}$, $\varphi$ is applied $2^j D$ times, and the square wave has a period of $2\pi/2^{j-1}$ ($D$ is the number of phases $\theta_i$, which determines the truncation order of the Fourier series of the square wave). By doubling the number of times $\varphi$ is applied in each iteration, the period of the square wave is progressively halved, which corresponds to halving the range of values within which $\varphi$ can fall (hence, each iteration returns one bit of the binary expansion of $\varphi$). This process can in principle be repeated indefinitely, allowing $\varphi$ to be estimated with arbitrary precision. The authors demonstrate that this iterative procedure reaches the Heisenberg limit, i.e., the uncertainty with which $\varphi$ is estimated is proportional to $1/N_p$, where $N_p$ is the number of times the phase $\varphi$ has to be applied in the algorithm.

This scheme has several benefits compared to alternative approaches for reaching the Heisenberg limit. First, as the authors mention, the Heisenberg limit is reached immediately, not asymptotically, as the estimate of $\varphi$ is not based on the output statistics. This is significant, as it means that Heisenberg-limited error scaling is obtained with finite resources. This contrasts with schemes that require the construction of an unbiased estimator using the measurement results, which in principle requires an infinite number of samples. Another benefit of this approach is that, unlike other adaptive strategies, this protocol does not require any prior information about $\varphi$, and there is no training or optimization involved. Typically, if the output probability distribution is periodic, multiple phases can lead to the same measurement outcome, which means that the phase cannot be uniquely determined without some prior knowledge. The protocol presented in Ref. [17] overcomes this by progressively halving the period of the square-wave probability distribution, as more information about the phase becomes available. The parameters $\theta_i$ that are used to control the response of the interferometer are known and fixed before the measurement process, so no prior optimization or real-time feedback is required for the control parameters. This makes the implementation of this approach simpler compared to adaptive strategies that require active feedback to adjust the control parameters during the measurement process (e.g., compare with [16]). Another key benefit of the proposal in Ref. [17] is that single-photon states are used as the probe, which are generally easier to generate than the multi-photon entangled states that are used in other proposals, like NOON states and squeezed states.

Summary and outlook

In summary, the paper by Roncallo et al. [17] presents a novel scheme for reaching Heisenberg-limited precision in phase estimation. The authors formulate the phase-estimation problem as a binary search strategy divided into multiple iterations, where the number of applications of the unknown phase $\varphi$ is progressively doubled in each iteration to estimate $\varphi$ with arbitrary precision, and the uncertainty is inversely proportional to the total number of times $\varphi$ is used. This work constitutes a significant step forward in optical quantum metrology, as the algorithm reaches the Heisenberg limit with finite resources and relatively simple input states (single photons), without the need for real-time feedback or prior parameter optimization.

At the end of this perspective, I would like to highlight some outstanding challenges related to the physical realization of this proposal, as potentially interesting considerations for future study. In practice, the unknown phase shift $\varphi$ corresponds to a physical object that photons have to pass through, which can result in photon loss due to absorption or scattering. If $p$ is the probability that a photon passes through the object once without getting lost, then after $N$ phase shifts the probability that the photon is still not lost decreases exponentially to $p^N$. For example, if $p = 0.8$ and $N = 10$, then there is a $0.8^{10} \approx 0.1$ probability that the photon hasn’t been lost, after 10 passes through the phase shift. As each subsequent iteration of the binary search requires twice as many uses of $\varphi$, photon loss is expected to limit the minimum uncertainty with which the phase can be estimated. A potential way to get around this is to repeat iterations where a photon is lost by sending in more photons, which would need to be included in the resource count of the algorithm. Another consideration is related to a comment made by the authors: If $\varphi$ cannot be applied at different spatial locations (e.g., if it is not possible to duplicate it), then a multi-round implementation would have to be adopted, where photons pass through the same phase shift multiple times. In this case, the control parameters $\theta_i$ would have to be changed in real time, and the number of times a photon passes through the unknown phase shift $\varphi$ would need to be progressively increased over time to go from one iteration to the next. This would potentially require reconfigurable photonic circuits with optical components such as single-photon switches that redirect photons through the interferometer, and send them to be measured at the output once all the necessary operations are applied. Such devices are at the forefront of research in nanophotonics [18,19,20] (accelerated by the promising applications of photonic quantum technologies [21]), suggesting that the proposal in Ref. [17] could be implemented in the near future.

Acknowledgments

I would like to thank Pieter Kok for valuable discussions and for helping with the preparation of this perspective. This work was supported by the Engineering and Physical Sciences Research council (Grant No. EP/W524360/1).

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