Adaptive Quantum Metrology under General Markovian Noise

We consider a general model of unitary parameter estimation in the presence of Markovian noise, where the parameter to be estimated is associated with the Hamiltonian part of the dynamics. In the absence of noise, the unitary parameter can be estimated with precision scaling as $1/T$, where $T$ is the total probing time. We provide a simple algebraic condition involving solely the operators appearing in the quantum master equation, implying, at most, $1/\sqrt{T}$ scaling of precision under the most general adaptive quantum estimation strategies. We also discuss the requirements a quantum error-correction-like protocol must satisfy in order to regain the $1/T$ precision scaling in case the above-mentioned algebraic condition is not satisfied. Furthermore, we apply the methods developed to understand fundamental precision limits in atomic interferometry with many-body effects taken into account, shedding new light on the performance of nonlinear metrological models.

Popular Summary

Quantum mechanics—the study of particles at the atomic scale—is widely perceived as a theory that prevents the precise measurement of physical quantities. The field of quantum metrology goes against this intuition, aiming to use quantum theory to make ultraprecise measurements. Gravitational-wave detectors, for example, aim to use quantum metrology to record extraordinarily subtle changes in the distance between a pair of mirrors in order to detect ripples in the fabric of spacetime. Noise from the environment, however, reduces the potential benefits of quantum metrology. We have developed a comprehensive theory for determining whether or not certain types of noise degrade measurements based on quantum metrology and, if so, whether it is possible to correct for this interference.

We consider the general problem of estimating a frequencylike parameter in the evolution of a quantum system interacting with its environment. Our approach allows for arbitrary quantum-enhanced techniques, such as entangled input probes and adaptive quantum operations, as well as the most general quantum measurements. We provide a simple algebraic condition that relates the noiseless part of the dynamics to the operators representing the environmental noise, which allows one to judge whether it is possible to achieve a quadratic improvement in precision scaling by using quantum protocols.

Our results give immediate identification of quantum systems in which the character of the noise allows, for example, the implementation of quantum error-correction codes to protect the sensing process from the impact of decoherence.

Figure 1

General adaptive quantum metrological scheme. Total evolution time $T$ is divided into a number $m$ of $t$-long steps interleaved with general unitary controls. The collective measurement performed in the end allows us to regard this scheme as a general adaptive protocol where measurement results at some stage of the protocol influence the control actions applied at later steps. This scheme may also mimic a parallel scheme where $m$ systems in an arbitrary initial entangled state are subject to evolution through $m$ parallel channels ${\mathcal{E}}_t^{\omega }$ for a time $t$.

Figure 2

Equivalent (up to linear order in $t$) representation of the $N$-particle dynamics in the form of a subsequent action of the $k$-particle Hamiltonian $H$ and $l$-particle noise $L$ on all $n$-particle subsets of the total $N$ particles (the circuit is given for $n=k=3\ge l=2$). Since the number of applications of the noise part is enhanced here by a factor ${\chi }_l\propto N^{n-l}$, we need to rescale the noise coefficient in the above scheme to ${\gamma }^{\prime }\propto \gamma /{\chi }_l$ to preserve the equivalence. Similarly, if $n>k$, we would need to rescale the frequency parameter to ${\omega }^{\prime }=\omega /{\chi }_k$. This representation allows us to calculate the bound on QFI for the whole dynamics by analyzing the properties of an elementary $n$-particle subchannel ${ϵ}_t^{{\omega }^{\prime },{\gamma }^{\prime }}$.