Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics

Abstract

This paper deals with the problem of identifying and estimating dynamical parameters of continuous-time Markovian quantum open systems, in the input-output formalism. First, we characterise the space of identifiable parameters for ergodic dynamics, assuming full access to the output state for arbitrarily long times, and show that the equivalence classes of undistinguishable parameters are orbits of a Lie group acting on the space of dynamical parameters. Second, we define an information geometric structure on this space, including a principal bundle given by the action of the group, as well as a compatible connection, and a Riemannian metric based on the quantum Fisher information of the output. We compute the metric explicitly in terms of the Markov covariance of certain ?fluctuation operators? and relate it to the horizontal bundle of the connection. Third, we show that the system-output and reduced output state satisfy local asymptotic normality, i.e., they can be approximated by a Gaussian model consisting of coherent states of a multimode continuous variables system constructed from the Markov covariance ?data.? We illustrate the result by working out the details of the information geometry of a physically relevant two-level systemauthorsversionPeer reviewe

Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment

“…Moreover, while most of the literature on parameter estimation with continuous measurements focuses on the information gained from the continuous signal, the crucial part that makes our protocol able to recover Heisenberg scaling is the final strong measurement on the conditional state. A great deal of effort has been also devoted to studying the asymptotic properties of estimation via repeated/continuous measurements [76][77][78]. In this approach one is usually interested in performing a single run of the experiment and thus observes the system for a long time.…”

Noise-assisted and monitoring-enhanced quantum bath tagging

“…( 28) is equal to 1/2 allows to find κ c in Eq. (31). The same analysis for the case with ĉ = σx 2 leads to the following critical value…”

“…The role of such an extra environment is twofold: on one hand, the presence of A is used as a way to positively interfere with the S-E coupling in an effort to increase the distinguishability among the quantum trajectories associated with the two hypothesis of the problem; on the second hand, A is employed to set up an indirect, continuous monitoring of the evolution of S, hence allowing us to acquire information about E in real time and not just at the end of the interaction interval. Continuous monitoring of quantum systems [24,25] has indeed been proven useful in the context of quantum metrology: in particular several works have either discussed the fundamental statistical tools to assess the precision achievable in this framework [26][27][28][29][30][31][32][33], and presenting practical estimation strategies [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. The theoretical framework needed to assess hypothesis testing protocols has been put forward first by Tsang [50] and then by Kiilerich and Molmer [51].…”

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“…The role of continuous measurements is indeed twofold: on the one hand, as it happens classically, the measurement output is exploited to acquire information on the parameters characterizing the system; on the other, the act of measuring alters the state of the system itself, thus opening the possibility of dynamically prepare more sensitive quantum probes. Several works have been proposed in the literature, both discussing the fundamental statistical tools to assess the precision achievable in this framework [5][6][7][8][9][10][11][12], and presenting practical estimation strategies [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”