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The evaluation of the minimal evolution time between two distinguishable states of a system is important for assessing the maximal speed of quantum computers and communication channels. Lower bounds for this minimal time have been proposed for unitary dynamics. Here we show that it is possible to extend this concept to nonunitary processes, using an attainable lower bound that is connected to the quantum Fisher information for time estimation. This result is used to delimit the minimal evolution time for typical noisy channels.

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.

The implementation of weak-value amplification requires the pre-and post-selection of states of a quantum system, followed by the observation of the response of the meter, which interacts weakly with the system. Data acquisition from the meter is conditioned to successful post-selection events. Here we derive an optimal post-selection procedure for estimating the coupling constant between system and meter, and show that it leads both to weak-value amplification and to the saturation of the quantum Fisher information, under conditions fulfilled by all previously reported experiments on the amplification of weak signals. For most of the pre-selected states, full information on the coupling constant can be extracted from the meter data set alone, while for a small fraction of the space of pre-selected states, it must be obtained from the post-selection statistics.

The estimation of parameters characterizing dynamical processes is a central problem in science and technology. It concerns for instance the evaluation of the duration of some interaction, of the value of a coupling constant, or yet of a frequency in atomic spectroscopy. The estimation error changes with the number N of resources employed in the experiment (which could quantify, for instance, the number of probes or the probing energy). For independent probes, it scales as 1/ √ N-the standard limit-a consequence of the central-limit theorem. Quantum strategies, involving for instance entangled or squeezed states, may improve the precision, for noiseless processes, by an extra factor 1/ √ N, leading to the so-called Heisenberg limit. For noisy processes, an important question is if and when this improvement can be achieved. Here, we review and detail our recent proposal of a general framework for obtaining attainable and useful lower bounds for the ultimate limit of precision in noisy systems. We apply this bound to lossy optical interferometry and show that, independently of the initial states of the probes, it captures the main features of the transition, as N grows, from the 1/N to the 1/ √ N behavior.