Narrow Results

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form $S_x+S_p\ge 1+ln\pi$, as well as in ordinary quantum mechanics.

In this work, we present the information theoretic lengths given by Shannon and Fisher for the lowest-lying stationary state of a one-dimensional ionic crystal modeled by $2N+1$ equally spaced attractive and repulsive Dirac delta potentials. The entropic uncertainty relations related to position and momentum spaces are studied as a function of the number of ions and the distance between them. Our results show that the stability of the ionic crystal depends on the number of ions and distance between them. In particular, we show that the position Shannon entropy is always positive and increases as the lattice constant between ions grows, in contrast with the momentum Shannon entropy which decreases and becomes negative beyond a particular lattice constant value.

In this paper, we compare different measurement strategies for estimating coupling strength of a Hamiltonian. In particular, we estimate the coupling strength coefficient when the Hamiltonian couples the pointer (measuring device) momentum to an observable of the system. We compare the performance of the estimations via four different measurement strategies: measuring the position or the momentum of the pointer with or without postselection of the system state. Without postselection, it turns out that all the information of the coupling strength can be extracted by measuring the position of the pointer. As a contrast, no information of the coupling strength can be extracted if the momentum is measured without postselection. Hence, measuring the position is a better choice if postselection is not allowed. However, when postselection is employed, we find that no matter which observable is measured, one can always achieve a great amplification of the measurement outcome, which can be larger than the initial standard deviation of the pointer observable under certain circumstances, at an acceptable cost of losing a little accuracy for the estimation of the coupling strength. With postselection, the better choice of the observable to be measured depends on the characteristic of the measurement devices.

The laser field interacting with single trapped ion in a carrier excitation frame in the presence of an external field is investigated. Detailed analytical expressions are given, taking into account carrier excitation configurations. We study the evolution of the population inversion, fidelity, ion-field entanglement, Fisher information and the geometric phase. The results indicate the influence of the external field on the information quantities to describe some physical phenomena. The relation between the population inversion, fidelity, von Neumann entropy, Fisher information, and geometric phase are explored during the time evolution.

The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.

The bound state solutions of the Schrödinger equation (SE) under a deformed hyperbolic potential are used in this paper to investigate the correlation between the information content, such as Shannon entropies and Fisher information, and the variance of a quantum system in both momentum and position spaces. The variance was obtained from the expectation moments in conjugated spaces and used to get the uncertainty products, Fisher information products and Shannon entropic sums for different potential parameters. These information measures were observed to vary with the potential parameters and obey their lowest bound inequalities. An increase in the Fisher information leads to a lower uncertainty and less information while decreases in the Fisher information result in higher uncertainties. We proposed a relationship between the Shannon entropies, Fisher information and the variance where an increase in the Fisher information leads to a lower Shannon entropy or a lower spread of the probability distribution and vice versa. The numerical results obtained with the proposed relations agree with the most utilized method in the literature. This current work simplifies the calculation of the Shannon entropies notably for complex potential functions.

A Schrödinger equation is solved for a cotangent hyperbolic potential using the elegant parametric Nikiforov–Uvarov method. The results obtained are used to calculate the Fisher information for position space and momentum space as well as the Shannon entropy for position space and momentum space at the ground state. The energies of the cotangent potential are observed to be fully bounded for different quantum states. The Fisher information satisfied the Fisher relation known as Cramer–Rao inequality for the Fisher product. Its result also obeyed the Heisenberg principle as its alternative for uncertainty measurement. However, the Shannon entropy only satisfied the Bialynick-Birula, Mycielski inequality for higher values of the screening parameter. Its relation to the Heisenberg principle failed as both the position space and the momentum space are parallel with high entropy squeezing. This potential has not been reported for any system to the best of our understanding.

We analyze the effects of quantum correlations on the parameter precision in an interferometric configuration. As probe states, we consider a class of two-qubit states for which the analytical expression of the quantum interferometric power, quantifying the quantum correlations, is explicitly derived. Also, we give and analyze the local quantum Fisher information, which evaluates the sensitivity of the probe state to the phase shift, for some relevant local Hamiltonians. The discord-like quantum correlations based on the notion of quantum interferometric power are compared with the original quantum discord based on von Neumann entropy. We also examine the significance of quantum correlations in enhancing the precision of the phase estimation. Our study corroborates the recent series of investigations focusing on the role of quantum correlations other than entanglement on the efficiency of quantum metrology protocols.

In this paper, Fisher information for an accelerated system initially prepared in the X-state is discussed. An analytical solution, which consists of three parts: classical, the average over all pure states and a mixture of pure states, is derived for the general state and for Werner state. It is shown that the Unruh acceleration has a depleting effect on the Fisher information. This depletion depends on the degree of entanglement of the initial state settings. For the X-state, for some intervals of Unruh acceleration, the Fisher information remains constant, irrespective to the Unruh acceleration. In general, the possibility of estimating the state’s parameters decreases as the acceleration increases. However, the precision of estimation can be maximized for certain values of the Unruh acceleration. We also investigate the contribution of the different parts of the Fisher information on the dynamics of the total Fisher information.

In this paper, an accelerated two-qubit system initially prepared in maximum or partial entangled state, which interacts locally with white/color or white-color noises, is considered. Due to the acceleration process and the noise effect, the entanglement degraded. Therefore, the effect of noise strength, initial state settings and the acceleration on the survival entanglement are investigated by means of the concurrence. Moreover, the initial parameters that describe this system are estimated by using the quantum Fisher information, where two forms are considered, namely by using a single and two-qubit forms. It is shown that, by using the two-qubit form, the estimation degree of these parameters is larger than that displayed by using a single-qubit form.

We employ techniques from quantum estimation theory (QET) to estimate the Lorentz violation parameters in the (1+3)-dimensional flat spacetime. We obtain and discuss the expression of the quantum Fisher information (QFI) in terms of the Lorentz violation parameter ${\sigma }_0$ and the momentum $k$ of the created particles. We show that the maximum QFI is achieved for a specific momentum $k_{\mathrm{\text{max}}}$. We also find that the optimal precision of estimation of the Lorentz violation parameter is obtained near the Planck scale.

We will use Fisher information to properly analyze the quantum weak equivalence principle. We argue that gravitational waves will be partially reflected by superconductors. This will occur as the violation of the weak equivalence principle in Cooper pairs is larger than the surrounding ionic lattice. Such reflections of virtual gravitational waves by superconductors can produce a gravitational Casimir effect, which may be detected using currently available technology.

In this work, we study quantum metrology protocol in the context of relativistic quantum fields in curved spacetimes. In particular, we restrict our attention to the isotropic and anisotropic models. We investigate how the oscillatory anisotropy affects a quantum metrology protocol. Here, the oscillatory anisotropy is incorporated as a small gravitational disturbance. We calculate the Fisher information for both cosmological parameters $𝜖$ (expansion volume) and $\rho$ (expansion rate). We find that, unlike the isotropic case, the Fisher information of both the parameters $𝜖$ and $\rho$ are oscillating functions of the momentum $k$. The oscillatory spectra of Fisher information can theoretically be understood as results of the dependence of the Bogoliubov coefficients on the anisotropic oscillations. In principle, this framework can be used to infer about the underlying spacetime structure and examine the effects of the universe scale factor oscillations.

This paper considers optimal designs of step-stress accelerated life tests in which each lognormally-distributed test item is first run at low stress, and if it does not fail for a specified time, then it is run at high stress until a predetermined censoring time. It is assumed that a log-linear relation exists between the lognormal location parameter and stress, and that a cumulative exposure model for the effect of changing stress holds. The optimum stress change point minimizes the asymptotic variance of maximum likelihood estimator of a specified percentile at design stress. For selected values of the design parameters, the optimum plans are tabulated. Designs of high-to-low step-stress accelerated life tests (ALTs) in which each item is first run at high stress and then at low stress, and the optimality criterion of minimizing the generalized asymptotic variance of maximum likelihood estimators of model parameters, are also considered. The effects of the incorrect pre-estimates of the design parameters are investigated.

We discuss the signal estimation that can be finished on a signal buried in generalized Gaussian noise based on the quantized version provided by a summing array of threshold devices. In the estimation, the Fisher information contained in the array output about the input signal is investigated. We show that the Fisher information can be improved as the noise intensity increases in the summing array and that a noise with a thinner tail in its distribution can lead to a better improvement, i.e., the fatter tail in the noise distribution may neutralize the beneficial role of noise. These results prove that the phenomenon of stochastic resonance (SR) or supra-threshold stochastic resonance (SSR) exists based on Fisher information in the summing array of threshold devices for generalized Gaussian noise, and that the noise distribution has an effect on the efficacy of SR or SSR. These above results also extend the applicability of the nonlinear phenomenon of SR or SSR in signal estimation.

If data is binary, it is probable that the combination of a signal of interest plus a noise has been simplified by a thresholding mechanism, as in, e.g., a neuron firing mechanism. For identifying optimal signal or coding range of binary data, Fisher information is an attractive measure. A general formula allows the signal level or signal range producing the most information-rich data to be identified if the noise distribution is known. In this paper we study the information content of binary data resulting from threshold exceedance of a signal plus an arbitrary type of noise. For a specified parametric family of distributions a fixed proportion of exceedances is optimal for any combination of signal, threshold, and noise amplitude. If the ratio of noise to signal level is constant, Fisher information is unimodal for many noise distributions. The results extend to the case of a random signal and to inter-exceedance-interval data. The family of gamma noise distributions is used for illustration.

Fisher information is a measure to quantify information and estimate system-defining parameters. The scaling and uncertainty properties of this measure, linked with Shannon entropy, are useful to characterize signals through the Fisher–Shannon plane. In addition, several non-gaussian distributions have been exemplified, given that assuming gaussianity in evolving systems is unrealistic, and the derivation of distributions that addressed asymmetry and heavy–tails is more suitable. The latter has motivated studying Fisher information and the uncertainty principle for skew-gaussian random variables for this paper. We describe the skew-gaussian distribution effect on uncertainty principle, from which the Fisher information, the Shannon entropy power, and the Fisher divergence are derived. Results indicate that flexibility of skew-gaussian distribution with a shape parameter allows deriving explicit expressions of these measures and define a new Fisher–Shannon information plane. Performance of the proposed methodology is illustrated by numerical results and applications to condition factor time series.

We introduce a new inaccuracy measure in terms of Fisher information. The proposed information measure is referred to as Fisher-based inaccuracy information (FBII) measure. Next, we examine some properties of this information measure and specifically examine it for escort and equilibrium distributions. Further, we propose Bayes–Fisher-based inaccuracy information (BFBII) measure and examine its connection to Kullback–Leibler and chi-square divergence measures. Moreover, in three different optimization problems, we show that the harmonic-mixture distribution gives optimal information based on BFBII measure. Some examples of FBII measure and escort density related to skew-normal and Student-t distributions are also illustrated, and then are applied to fish condition factor time series.

Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e. the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.