Quantum Metrological Limits via a Variational Approach

Abstract

The minimum achievable statistical uncertainty in the estimation of physical parameters is determined by the quantum Fisher information. Its computation for noisy systems is still a challenging problem. Using a variational approach, we present an equation for obtaining the quantum Fisher information, which has an explicit dependence on the mathematical description of the noise. This method is applied to obtain a useful analytical bound to the quantum precision in the estimation of phase-shifts under phase diffusion, which shows that the estimation uncertainty cannot be smaller than a noise-dependent constant. Introduction.-Quantum metrology [1-4] deals with the estimation of parameters taking into account the constraints imposed by quantum laws. The estimation is based on measurements made on probe systems undergoing a parameter-dependent process. For a given measurement scheme, the uncertainty in the estimation of a parameter is limited by the Cramér-Rao bound, which is proportional to the inverse of the square root of the so-called Fisher information (FI) [5][6][7]. The maximization of FI over all measurement strategies allowed by quantum mechanics leads to a non-trivial quantity: the quantum Fisher information (QFI). The determination of this quantity is central to quantum metrology. It allows, for instance, the establishment of ideal benchmarks for the statistical uncertainty in the estimation of parameters, which can be used by experimentalists to evaluate the performance of a real experiment. A systematic approach to calculate the QFI, using the symmetric logarithmic derivative (SLD) operator [1,2], was developed in Ref. [3]. This approach has allowed large advances on quantum metrology [8,9]. For unitary processes, it leads to simple analytical expressions. This is not the case, however, for noisy processes, which often require numerical calculations.

Precision bounds for noisy nonlinear quantum metrology

“…In the following, we address this question using the variational approach to the method of purifications introduced in Ref. [18].…”

“…In Ref. [18] it was shown that the ultimate precision bound for a linear estimation scheme affected by phase diffusion contains a constant term independent of the average number of photons N . Hence, it is believed that the presence of phase diffusion is even more detrimental to the performance of linear estimation than the presence of photon loss.…”

“…where β denotes the strength of phase-diffusion noise [18] and |θ E denotes the initial Gaussian state of the environment which in the position eigenbasis reads…”

“…So far the analysis of noisy measurement schemes was limited only to schemes governed by linear generators leading to some very pessimistic results [12][13][14][15][16][17][18][19]. Although, under certain circumstances such as the presence of non-Markovian environments [20,21], or frequency estimation with transversal noise [22], some improvement in the precision with respect to the shot-noise limit can be observed.…”

“…In our study, we derive the ultimate bounds on the performance of noisy nonlinear measurement schemes and determine whether such schemes can offer any improvement with respect to their noisy linear counterparts. In particular, we investigate the performance of the second-order estimation scheme in the presence of the two most common types of noise, photon loss and phase diffusion, using a number of different techniques [16,18,23,24]. While photon loss is typically considered the most dominant source of noise for optical quantum systems, in atomic clocks, for example, phase fluctuations may be more important [25].…”