On the calculation of Fisher information for quantum parameter estimation based on the stochastic master equation

The Fisher information can be used to indicate the precision of parameter estimation by the quantum Cramer-Rao inequality. This paper presents an efficient numerical algorithm for the calculation of Fisher information based on quantum weak measurement. According to the quantum stochastic master equation, the Fisher information is expressed in the form of log-likelihood functions. Three main methods are employed in this algorithm: (i) we use the numerical differentiation approach to calculate the derivative of the log-likelihood function; (ii) we randomly generate a series of parameters of interest by the Metropolis Hastings (MH) algorithm; and (iii) the values of expectation can be approximated by the Markov chain Monte Carlo (MCMC) integration. Finally, as an example to testify the feasibility of the proposed algorithm, we consider the dissipation rates of the open quantum system as unknown parameters that need to be estimated. We show that the Fisher information can reach a precision close to the Heisenberg limit in the weak coupling condition. This again demonstrates the effectiveness of the new algorithm.

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