Optimal Measurement Angle Research Articles

Optimizing experimental design in neutron reflectometry.

Using the Fisher information (FI), the design of neutron reflectometry experiments can be optimized, leading to greater confidence in parameters of interest and better use of experimental time [Durant, Wilkins, Butler & Cooper (2021). J. Appl. Cryst. 54, 1100-1110]. In this work, the FI is utilized in optimizing the design of a wide range of reflectometry experiments. Two lipid bilayer systems are investigated to determine the optimal choice of measurement angles and liquid contrasts, in addition to the ratio of the total counting time that should be spent measuring each condition. The reduction in parameter uncertainties with the addition of underlayers to these systems is then quantified, using the FI, and validated through the use of experiment simulation and Bayesian sampling methods. For a 'one-shot' measurement of a degrading lipid monolayer, it is shown that the common practice of measuring null-reflecting water is indeed optimal, but that the optimal measurement angle is dependent on the deuteration state of the monolayer. Finally, the framework is used to demonstrate the feasibility of measuring magnetic signals as small as 0.01 μB per atom in layers only 20 Å thick, given the appropriate experimental design, and that the time to reach a given level of confidence in the small magnetic moment is quantifiable.

Landau-type sudden transitions of quantum correlations

Sudden changes of quantum correlations in the Bell-diagonal states are well-known effects. They occur when the set of optimal parameters that determine the quantum correlation consists of isolated points and optimal parameters during the evolution of the system jump from one such point to another (e.g., the optimal measurement angle of the quantum discord changes discontinuously from zero to π/2 or vice versa). However, when considering more general X quantum states, we found that quantum discord and one-way quantum work deficit can experience sudden changes of other kinds. Namely, the optimal measurement angle may suddenly start to shift continuously from its stationary value 0 or π/2 to an intermediate optimal measurement angle ϑ ∈ (0, π/2). This leads to a new behavior of quantum correlations, which is mathematically described by the Landau phenomenological theory of second-order phase transitions. In addition, for the one-way quantum work deficit, we found cases where the optimal measurement angle jumps from zero to a nonzero step less than π/2, and then continuously changes its value. This behavior of quantum correlation is similar to a first-order phase transition in Landau’s theory. Dependencies of quantum discord and one-way quantum work deficit near the boundaries, which separate regions with state-dependent (variable) and state-independent (stationary, constant) optimal measurement angles, are examined in detail on an example of the XXZ spin model in an external field at thermal equilibrium.

Temperature-field phase diagrams of one-way quantum work deficit in two-qubit XXZ spin systems

The spin-1/2 XXZ chain in a uniform magnetic field at thermal equilibrium is considered. For this model, we give a complete classification of all qualitatively different phase diagrams for the one-way quantum work (information) deficit. The diagrams can contain regions (phases, fractions) with both stationary and variable (state-dependent) angles of optimal measurement. We found cases of phase diagrams in which the sizes of regions with the variable optimal measurement angle are large and perhaps such regions can be detected experimentally. We also established a relationship between the behavior of optimal measurement angles near the boundaries separated different regions and Landau's theory of phase transitions of the second and first kind.

On the possibility to detect quantum correlation regions with the variable optimal measurement angle

Quantum correlations described by quantum discord and one-way quantum deficit can contain ordinary regions with {\em constant} (i.e., universal) optimal measurement angle $0$ or $\pi/2$ with respect to the $z$-axis and regions with a {\em variable} (state-dependent) angle of the optimal measurement. The latter regions which are absent in the Bell-diagonal states are very tiny for the quantum discord and cannot be observed experimentally due to various imperfections on the preparation and measurement steps of the experiment. On the contrary, for the one-way quantum deficit we succeeded in getting the special two-qubit X states which seem to allow one to reach all regions of quantum correlation exploiting available quantum optical techniques. These states give possibility to deep investigation of quantum correlations and related optimization problems at new region and its boundaries. In the paper, explicit theoretical calculations applicable to one-way deficit are reported, together with the design of the experimental setup for generating such selected family of states; moreover, there are presented numerical simulations showing that the most inaccessible region with the intermediate optimal measurement angle may be resolved experimentally.

Phase diagram for the one-way quantum deficit of two-qubit X states

The one-way quantum deficit, a measure of quantum correlation, can exhibit for X quantum states the regions (subdomains) with the phases $\Delta_0$ and $\Delta_{\pi/2}$ which are characterized by constant (i.e., universal) optimal measurement angles, correspondingly, zero and $\pi/2$ with respect to the $z$-axis and a third phase $\Delta_\vartheta$ with the variable (state-dependent) optimal measurement angle $\vartheta$. We build the complete phase diagram of one-way quantum deficit for the XXZ subclass of symmetric X states. In contrast to the quantum discord where the region for the phase with variable optimal measurement angle is very tiny (more exactly, it is a very thin layer), the similar region $\Delta_\vartheta$ is large and achieves the sizes comparable to those of regions $\Delta_0$ and $\Delta_{\pi/2}$. This instils hope to detect the mysterious fraction of quantum correlation with the variable optimal measurement angle experimentally.

Determination of Aerosol Size Distribution from Angular Light-Scattering Signals by Using a SPSO-DE Hybrid Algorithm

The comparison of the angular light-scattering method (ALSM) and the spectral extinction method (SEM) in solving the inverse problem of aerosol size distribution (ASD) are studied. The inverse problem is solved by a SPSO-DE hybrid algorithm, which is based on the stochastic particle swarm optimization (SPSO) algorithm and differential evolution (DE) algorithm. To improve the retrieval accuracy, the sensitivity analysis of measurement signals to characteristic parameters in ASDs is studied; and the corresponding optimal measurement angle selection region for ALSM and optimal measurement wavelength selection region for SEM are proposed, respectively. Results show that more satisfactory convergence properties can be obtained by using the SPSO-DE hybrid algorithm. Moreover, short measurement wavelengths and forward measurement angles are beneficial to obtaining more accurate results. Then, common monomodal and bimodal ASDs are estimated under different random measurement errors by using ALSM and SEM, respectively. Numerical tests show that retrieval results by using ALSM show better convergence accuracy and robustness than those by using SEM, which is attributed to the distribution of the objective function value. As a whole, considering the convergence properties and the independence on prior optical information, the ALSM combined with SPSO-DE hybrid algorithm provides a more effective and reliable technique to obtain the ASDs.

Retrieval of aerosol optical constants from angular light-scattering measurement signals using principal component analysis approach

An angular light-scattering measurement (ALSM) method combined with the probability density function-based ant colony optimization algorithm (PDF-ACO) is proposed for retrieval of aerosol optical constants. An optimal measurement angle selection method using a principal component analysis (PCA) approach is developed to improve retrieval accuracy. Results indicate that optimized angle selection can ensure retrieval accuracy. The aerosol optical constants over Beijing, China, which are available from the Aerosol Robotic Network (AERONET), are then reconstructed. The ALSM method’s convergence properties are also studied via comparison with those of the light reflection-transmittance measurement (LRTM) method. Results retrieved using the ALSM method show better convergence speed and accuracy than those retrieved using the LRTM method because the ALSM method does not require solution of the radiative transfer equation and allows more useful signals to be obtained. Additionally, the inverse accuracy of the refractive index results is better than that of the absorption index results; this is attributed to differences between the monodromic characteristics of the refractive index and absorption index retrieval results. All results confirm that the combination of the ALSM method with the PDF-ACO algorithm and the optimal measurement angle selection method provides effective and reliable aerosol optical constant reconstruction.

Extremal properties of conditional entropy and quantum discord for XXZ, symmetric quantum states

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy $$S_{cond}$$ as a function of measurement angle $$\theta \in [0,\pi /2]$$ . Numerical calculations show that the function $$S_{cond}(\theta )$$ for X states can have at most one local extremum in the open interval from zero to $$\pi /2$$ (unimodality property). If the extremum is a minimum, the quantum discord displays region with variable (state-dependent) optimal measurement angle $$\theta ^*$$ . Such $$\theta $$ -regions (phases, fractions) are very tiny in the space of X-state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval $$(0,\pi /2)$$ . It is remarkable that the maxima exist in surprisingly wide regions, and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum.

A novel multi-probe method for separating spindle radial error from artifact roundness error

Conventional three-probe method of artifact roundness and spindle error measurements is subjected to the trouble of harmonic suppression, which is also mathematically complicated due to complex transformation processing. A novel three-probe method by solving system of multivariable equation (SSME) method is presented in this paper. The presented method simplifies the mathematical processing and has good robustness to the measurement angles which are difficult for conventional three-probe method. This paper gives mathematical models, detailed theoretical derivations, and numerical simulations. Selection criteria of the optimal measurement angle combinations can be determined in accordance with rank and condition number of the coefficient matrix. Then, to validate the feasibility and repeatability of the proposed method, experimental measurements are performed on a vertical machine tool spindle by using nanometer-resolution capacitive displacement sensors, high-precision indexing table, and professional data acquisition system. Artifact roundness error and spindle radial error are tested to validate separation accuracy by specialized instruments. The comparisons demonstrate that the proposed method has good feasibility and repeatability (the maximum deviations of spindle error and roundness error are 9.62 and 3.86%). SSME method, due to its simplicity in computation and the uniqueness of the solution provided, is more suitable to separate spindle radial error from the artifact roundness error.

On the quantum discord of general X states

Quantum discord Q is a function of density matrix elements. The domain of such a function in the case of two-qubit system with X density matrix may consist of three subdomains at most: two ones where the quantum discord is expressed in closed analytical forms (Q_{\pi/2} and Q_0) and an intermediate subdomain for which, to extract the quantum discord Q_\theta, it is required to solve in general numerically a one-dimensional minimization problem to find the optimal measurement angle \theta\in(0,\pi/2). Hence the quantum discord is given by a piecewise-analytic-numerical formula Q=\min{Q_{\pi/2}, Q_\theta, Q_0}. Equations for determining the boundaries between these subdomains are obtained. The boundaries consist of bifurcation points. The Q_{\theta} subdomains are discovered in the generalized Horodecki states, in the dynamical phase flip channel model, in the anisotropic spin systems at thermal equilibrium, in the heteronuclear dimers in an external magnetic field. We found that transitions between Q_{\theta} subdomain and Q_{\pi/2} and Q_0 ones occur suddenly but continuously and smoothly, i.e., nonanalyticity is hidden and can be observed in higher derivatives of discord function.

NMR dynamics of quantum discord for spin-carrying gas molecules in a closed nanopore

A local orthogonal transformation that transforms any centrosymmetric (CS) matrix of fourth order to the X form is found. A picewise-analytic-numerical formula Q=min{Q_\pi/2,Q_\theta,Q_0}, where Q_\pi/2 and Q_0 are analytical expressions and the branch Q_{\theta} is found by numerical searching the optimal measurement angle \theta\in(0,\pi/2), is proposed to calculate the quantum discord Q of general X state. The developed approaches are applied to a quantitative description of recently predicted flickerings (periodic birth and death) of the quantum-information pair correlation between nuclear 1/2 spin of atoms or molecules of a gas (for examples Xe^129) in a finite asymmetric volume in the presence of a strong magnetic field.