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 Cramér-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. Introduction In reality, not all parameters in the open quantum system1, 2 can be obtained directly. Moreover, there are always some inevitable estimation errors in those who need to be estimated. How to reduce the errors has become a key problem in recent years. The parameter estimation theory3–5 tells us that any parameter estimation has an estimated precision and it’s hard to surpass the limit precision by the traditional methods. In the classical case, the maximum precision is called the standard quantum limit6 or shot noise limit, 1/ √ N, where N represents the number of experiments or that of general particle experiments. However, Caves7, 8 showed that with the help of squeezed state technique, quantum mechanical systems can achieve greater sensitivity over the standard quantum limit. Theoretically the ultimate precision limit is the Heisenberg limit9 1/N. The classic Fisher information was originally used to describe the information of unknown parameter contained in a random variable. It’s well-known that the variance of any unbiased estimation is at least as high as the inverse of the Fisher information10–13 . This is also known as the Cramér-Rao inequality , Varθ̂ ≥ 1/I (θ). The Fisher information provides a better way to calculate the estimation precision. Since quantum projective measurements will in general disturb the system they are measuring, the Fisher information may undergoes a large deviation. Improving the accuracy of quantum parameter estimation based on quantum weak measurement14–16 caused a wide range of interests. Smith and co-authors proposed a protocol to achieve fast, accurate and non-destructive quantum state estimation based on continuous weak measurement in the presence of a controlled dynamical evolution17 . Xu and co-authors used a weak measurement scheme to realize high precision quantum phase estimation18 . Gammelmark and Mølmer derived a likelihood function to estimate the unknown parameters with the help of quantum measurement and quantum stochastic master equation19 . They investigated the statistical properties of the output state, which will provide the ultimate limits in estimation precision. Although much progress has been made in quantum parameter estimation based on continue weak measurement, how to effectively calculate the Fisher information based on the quantum stochastic master equation is still with remarkable difficulty. To figure out this problem, one needs to represent the Fisher information in computable forms and take effective measures to prior-estimate the parameter of interest. Recently, Genoni proposed a method to calculate the Fisher information for linear Gaussian quantum system, whose evolution depends only on the evolution of first and second moments of the quantum states20 . In this paper, we propose an efficient numerical algorithm to calculate the Fisher information based on the quantum stochastic master equation. 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) algorithm21 ; and (iii) the values of expectation can be approximated by the Markov chain Monte ar X iv :1 70 2. 08 08 9v 2 [ qu an tph ] 2 M ar 2 01 7 Carlo (MCMC) integration22 . Moreover, we reduce the complexity of calculation by showing that any quantum stochastic master equation for un-normalized states corresponds to the evolutions of some normalized states. The result of this work opens an efficient way to obtain the ultimate precision of parameter estimation for the open quantum system. Results Stochastic master equation based on quantum weak measurement We consider the quantum parameter estimation of the open quantum system based on quantum weak measurement. The unknown parameters may exist in the system Hamiltonian, the dissipation rates, and coupling or measurement strength. Here, the measurement process is assumed to be a Markov process. During the measurement and estimation processes, the quantum stochastic master equation method has always been involved. For brevity, the stochastic master equation23–25 based on quantum weak measurement for an un-normalized state ρ̃t is given by dρ̃t =−i [H, ρ̃t ]dt + ( Lρ̃tL− 1 2 ( LLρ̃t + ρ̃tLL )) dt + √ η ( Lρ̃t + ρ̃tL ) dYt , (1) where H is the Hamiltonian of the quantum system, η is the measurement strength with the weak measurement constraint η 1, and dYt is the infinitesimal increment which represents the measurement output. Based on the relationship between an un-normalized quantum state ρ̃t and a normalised state ρt , ρt=ρ̃t/Tr(ρ̃t), we have dYt = √ ηTr ( ρtL+Lρt ) dt +dWt , (2) where dWt is the Winner increment with zero mean and variance dt. It describes the quantum fluctuations of the continuous output signal. For convince, we introduce a map M (ρ) = Lρ +ρL†, and a likelihood function Lt = Tr(ρ̃t). Below, we derive a log-likelihood function which is closely related to Fisher information and stands for the precision of parameter estimation. According to Eq. (1), the derivative of the likelihood function Lt with respect to time t can be written as dLt = Tr(dρ̃t) = √ ηTr(M (ρ̃t))dYt = √ ηTr(M (ρt))LtdYt . (3) Thus, we can obtain the normalized quantum stochastic master equation by means of the multi-dimensional Itô formula. dρt =−i [H,ρt ]dt + ( LρtL− 1 2 ( LLρt +ρtLL )) dt + √ η (M (ρt)−ρtTr(M (ρt)))dWt . (4) Quantum Fisher information Suppose θ is an unknown parameter of the open quantum system that needs to be estimated. As mentioned above, the Fisher information can be used to indicate the precision of parameter estimation by the quantum Cramér-Rao inequality26–28 , i.e, 〈( δ θ̂ )2〉≥ 1 NI (θ) , (5) where I (θ) is the Fisher information and N is the number of measurements. Obviously, if I (θ) approaches N, it means that the estimation precision is closed to the Heisenberg limit. Below, we use lt to denote the log-likelihood function29, 30 , and we have dlt = d lnLt = dLt Lt = √ ηTr(M (ρt))dYt . (6) Therefore, the Fisher information can be rewritten as I (θ) = E [( ∂ lnLt ∂θ )2] = E [( ∂ lt ∂θ )2] . (7) Substituting Eq. (6) into Eq. (7), we can obtain the analytic form of the quantum Fisher information. Calculation of the quantum Fisher information From the Fisher information Eq. (7), it is easy to find that θ is not an independent variable of the likelihood function. In other words, it is not an explicit expression and that makes the calculation with remarkable difficulty. In order to efficiently calculate the quantum Fisher information, we propose a numerical algorithm with the help of MH algorithm and MCMC integration.