Parameter estimation for quantum jump unraveling

TL;DR

This work tackles parameter estimation from continuous quantum-jump records where temporal correlations challenge standard Fisher Information analysis. It develops explicit FI expressions for multi-channel renewal processes in terms of waiting-time distributions and introduces a Fisher-Gillespie algorithm to efficiently compute FI along non-renewal trajectories, complemented by a monitoring-operator approach to enable stable MLE estimation. It also analyzes information loss under data compression and post-selection, providing bounds and decomposition of FI into channel and timing contributions. Together, these results furnish a practical toolbox for quantum metrology in jump unravelings, with implications for designing high-precision quantum sensors in quantum optics and condensed-matter platforms.

Abstract

We consider the estimation of parameters encoded in the measurement record of a continuously monitored quantum system in the jump unraveling, corresponding to a single-shot scenario, where information is continuously gathered. Here, it is generally difficult to assess the precision of the estimation procedure via the Fisher Information due to intricate temporal correlations and memory effects. In this paper we provide a full set of solutions to this problem. First, for multi-channel renewal processes we relate the Fisher Information to an underlying Markov chain and derive a easily computable expression for it. For non-renewal processes, we introduce a new algorithm that combines two methods: the monitoring operator method for metrology and the Gillespie algorithm which allows for efficient sampling of a stochastic form of the Fisher Information along individual quantum trajectories. We show that this stochastic Fisher Information satisfies useful properties related to estimation on a single run. Finally, we consider the case where some information is lost in data compression/post-selection and provide tools for computing the Fisher Information in this case. All scenarios are illustrated with instructive examples from quantum optics and condensed matter.

Figures (10)

• Figure 1: A quantum process emits a signal every time a jump occurs. There are two different jump channels (labeled as $L_+$ and $L_-$), corresponding to distinguishable signals. The experimenter has access only to the measurement record, constituted by jump time and jump channel for each emission. In the illustration, the quantum system is a qubit coupled to a thermal bath, whose behavior is discussed in detail in Sec. \ref{['sec:qubit_thermometry']}. The plot illustrates the expectation value of the number observable for the system as a function of time.
• Figure 2: Estimation task results for the coupled qubit model. MLE estimation on 500 trajectories for a fixed final time 100 (a) yields normally distributed estimates; the variance is compatible with the prediction of the Cramér-Rao bound. (b): the variance of the MLE estimates is compared with the Cramér-Rao bound. As expected, perfect saturation is not reached for finite time, but the error on the estimates is comparable in order of magnitude with the bound. Parameters: $\Omega_A = \Omega_B = \bar{n} = 1$, $\gamma = 0.4$, $g = 0.01$, timestep $dt = 0.001$, $\gamma$ increment for the derivatives $d\gamma = 0.0001$.
• Figure 3: Thermometry with a qubit with observed raising and lowering operations. (a) The WTD for different observations. Parameters: $\omega=\Omega=\gamma=1$ and $\bar{n}=1.5$. (b) The Fisher information per observed jumps. Parameters: $\omega=\Omega=\gamma=1$. The total rate (blue) is shown with the contribution from the channels $F^{\rm ch}$ alone (orange). (c) Proportion of total Fisher information in the channels, alone, for different Rabi oscillations $\Omega$ and $\omega=\gamma=1$. (d) The Fisher information per observed jumps as a function of $\Omega$, same parameters as in (b).
• Figure 4: Fisher information of the entire measurement record, and of the sample mean for the resonant fluorescence process. In the main panel, the Fisher information rate for the Rabi frequency $\Omega = 1$ is represented as a function of the jump rate $\Gamma$; in the inset plot, for fixed $\Gamma = \Omega = 1$, as a function of $N$, the number of recorded jumps.
• Figure 5: A depiction of the coupled-qubit model.

Theorems & Definitions (3)

• Lemma 1
• proof
• Lemma 2