Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment

Abstract

We study quantum frequency estimation for N qubits subject to independent Markovian noise via strategies based on time-continuous monitoring of the environment. Both physical intuition and the extended convexity of the quantum Fisher information (QFI) suggest that these strategies are more effective than the standard ones based on the measurement of the unconditional state after the noisy evolution. Here we focus on initial GHZ states subject to parallel or transverse noise. For parallel, i.e., dephasing noise, we show that perfectly efficient timecontinuous photodetection allows us to recover the unitary (noiseless) QFI, and hence obtain Heisenberg scaling for every value of the monitoring time. For finite detection efficiency, one falls back to noisy standard quantum limit scaling, but with a constant enhancement due to an effective reduced dephasing. In the transverse noise case, Heisenberg scaling is recovered for perfectly efficient detectors, and we find that both homodyne and photodetection-based strategies are optimal. For finite detector efficiency, our numerical simulations show that, as expected, an enhancement can be observed, but we cannot give any conclusive statement regarding the scaling. We finally describe in detail the stable and compact numerical algorithm that we have developed in order to evaluate the precision of such time-continuous estimation strategies, and that may find application in other quantum metrology schemes.Accepted in Quantum 2018-11-29, click title to verify arXiv:1803.05891v3 [quant-ph]

A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging

“…Naively, monitoring additional nuisance parameters may improve the stability of the estimation, a task which is relevant for moving towards real-life devices. In this respect, one should also consider the performance of sensors based on time-continous monitoring; the corresponding mathematical framework needed to assess their optimality in terms of multi-parameter quantum estimation has already been established [176,177], and single-parameter theoretical results have been recently obtained in different contexts [178][179][180]. We also remark that for continuous monitoring the classical data processing involved in estimation is highly non-trivial, see e.g.…”

Quantum frequency estimation with conditional states of continuously monitored independent dephasing channels

“…In general, obtaining an expression for the state |Ψ SE is a hard task, but there is a method based on a modified master equation to compute the fidelity Ψ SE (ω 1 )|Ψ SE (ω 2 ) without obtaining the state [21,54,55]. This technique is fundamental to obtain the results for frequency estimation with continuous monitoring of observables transversal to the Hamiltonian [29]. However, in this paper we deal with collapse operators that commute with the Hamiltonian and calculations are greatly simplified.…”

“…However, in this paper we deal with collapse operators that commute with the Hamiltonian and calculations are greatly simplified. As a matter of fact, whenever the collapse operators commute with the HamiltonianĤ ω , the ultimate QFI Q Lω,κ is equal to the QFI of the state evolving under the unitary dynamics generated byĤ ω , as shown in [29].…”

“…where the summation is over all the possible trajectories that can followed by the system. This quantity can be computed by simulating the stochastic master equation, together with an additional master equation as shown in [20], see also [29] for a stable implementation of this method that takes advantage of the Kraus operator form of the infinitesimal evolution introduced in [56]. We also mention that for Gaussian systems, Gaussian measurements and Gaussian dynamics, the computation can be performed more effectively in the phase-space picture [26].…”

“…In this work we focus on Markovian dephasing noise only and we expand on the results of [29], presenting considerably more general results, that are valid for arbitrary initial states. Furthermore, we now present a fully analytical solution of the dynamics for generic efficiency of the monitoring, for both continuous photodetection and homodyne detection, while previously the case of inefficient detection was treated numerically.…”