Dynamics of measurement-induced non-locality and geometric discord with initial system-environment correlations

We analyze the dynamics of geometric measure of discord （GMOD） and measurement-induced non-locality （MIN） in the presence of initial system-reservoir correlations without Born and Markov approximation. Although the initial system-environment states have the same reduced density matrices for both the system and environment, the effects of different initial system-environment correlations have been shown to fundamentally alter the time evolution of GMOD and MIN between two quantum systems in both Markovian and non-Markovian regimes. In general, both GMOD and MIN experience a sudden increase for initially quantum-correlated states, and a sudden decrease for classical-correlated states before they reach the same stationary values with initially factorized states.

A symmetric geometric measure and the dynamics of quantum discord

A symmetric measure of quantum correlation based on the Hilbert–Schmidt distance is presented in this paper. For two-qubit states, we considerably simplify the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering from dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There exists a certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.

Revised Geometric Measure of Entanglement for Multipartite and Continuous Variable Systems

Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state （GSS） in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement （GME） and relative entropy of entanglement （RE）. The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.

MEASUREMENT OF ANGULAR VIBRATION AMPLITUDE BY ACTIVELY BLURRED IMAGES

A novel motion-blur-based method for measuring the angular amplitude of a high-frequency rotational vibration is schemed. The proposed approach combines the active vision concept and the mechanism of motion-from-blur, generates motion blur on the image plane actively by extending exposure time, and utilizes the motion blur information in polar images to estimate the angular amplitude of a high-frequency rotational vibration. This method obtains the analytical results of the angular vibration amplitude from the geometric moments of a motion blurred polar image and an unblurred image for reference. Experimental results are provided to validate the presented scheme.

ON STATIONARY HYPERSURFACES IN EUCLIDEAN SPACES

Some techniques in the Geometric Measure Theory are used to study the hypersurfaces in Euclidean spaces and some fundamental properties with this subject are discussed in this article.

Uncertainty analysis of measured geometric variations in turbine blades and impact on aerodynamic performance

Inevitable geometric variations significantly affect the performance of turbines or even that of entire engines;thus,it is necessary to determine their actual characteristics and accurately estimate their impact on performance.In this study,based on 1781 measured profiles of a typical turbine blade,the statistical characteristics of the geometric variations and the uncertainty impact are analyzed,and some commonly used uncertainty modelling methods based on Principal-Component Analysis(PCA)are verified.The geometric variations are found to be evident,asymmetric,and non-uniform,and the non-normality of the random distributions is non-negligible.The performance is notably affected,which is manifested as an overall offset,a notable scattering,and significant deterioration in several extreme cases.Additionally,it is demonstrated that the PCA reconstruction model is effective in characterizing major uncertainty characteristics of the geometric variations and their impact on the performance with almost the first 10 PCA modes.Based on a reasonable profile error and mean geometric deviation,the Gaussian assumption and stochasticprocess-based model are also found to be effective in predicting the mean values and standard deviations of the performance variations.However,they fail to predict the probability of some extreme cases with high loss.Finally,a Chi-square-based correction model is proposed to compensate for this deficiency.The present work can provide a useful reference for uncertainty analysis of the impact of geometric variations,and the corresponding uncertainty design of turbine blades.

THE GAUSS–GREEN THEOREM IN CLIFFORD ANALYSIS AND ITS APPLICATIONS

In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.

Quantum correlation switches for dipole arrays

We investigate the characteristics of three kinds of quantum correlations, measured by pairwise quantum discord （QD）, geometric measure of quantum discord （GMQD）, and measurement-induced disturbance （MID）, in the systems of three- and four-dipole arrays. The influence of the temperature on the three quantum correlations and entanglement of the systems is also analyzed numerically. It is found that novel quantum correlation switches called QD, GMQD, and MID respectively can be constructed with the qubits consisting of electric dipoles coupled by the dipole-dipole interaction and oriented along or against the external electric field. Moreover, with the increase of temperature, QD, GMQD, and MID are more robust than entanglement against the thermal environment. It is also found that for each dipole pair of the three- and four-dipole arrangements, the MID is always the largest and the GMQD the smallest.

A graph autoencoder network to measure the geometric similarity of drainage networks in scaling transformation

Similarity measurement has been a prevailing research topic geographic information science.Geometric similarity measurement inin scaling transformation(GSM_ST)is critical to ensure spatial data quality while balancing detailed information with distinctive features.However,GSM_ST is an uncertain problem due to subjective spatial cognition,global and local concerns,and geometric complexity.Traditional rule-based methods considering multiple consistent conditions require subjective adjustments to characteristics and weights,leading to poor robustness in addressing GSM_ST.This study proposes an unsupervised representation learning framework for automated GSM_ST,using a Graph Autoencoder Network(GAE)and drainage networks as an example.The framework involves constructing a drainage graph,designing the GAE architecture for GSM_ST,and using Cosine similarity to measure similarity based on the GAE-derived drainage embeddings in different scales.We perform extensive experiments and compare methods across 71 drainage networks duringfive scaling transformations.The results show that the proposed GAE method outperforms other methods with a satisfaction ratio of around 88%and has strong robustness.Moreover,our proposed method also can be applied to other scenarios,such as measuring similarity between geographical entities at different times and data from different datasets.

quantum discord geometric measure quantum phase transition

A generalization of the geometric measure of quantum discord is introduced in this article, based on Hellinger distance. Our definition has virtues of computability and independence of local measurement. In addition it also does not suffer from the recently raised critiques about quantum discord. The exact result can be obtained for bipartite pure states with arbitrary levels, which is completely determined by the Schmidt decomposition. For bipartite mixed states the exact result can also be found for a special case. Furthermore the generalization into multipartite case is direct. It is shown that it can be evaluated exactly when the measured state is invariant under permutation or translation. In addition the detection of quantum phase transition is also discussed for Lipkin–Meshkov–Glick and Dicke model.

Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique

The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.

Geometric measure of quantum discord and the geometry of a class of two-qubit states

We investigate the geometric picture of the level surfaces of quantum entanglement and geometric measure of quantum discord(GMQD) of a class of X-states, respectively. This pictorial approach provides us a direct understanding of the structure of entanglement and GMQD. The dynamic evolution of GMQD under two typical kinds of quantum decoherence channels is also investigated. It is shown that there exists a class of initial states for which the GMQD is not destroyed by decoherence in a finite time interval. Furthermore, we establish a factorization law between the initial and final GMQD, which allows us to infer the evolution of entanglement under the influences of the environment.

Dynamics of coherence-induced state ordering under Markovian channels

We study the dynamics of coherence-induced state ordering under incoherent channels, particularly four specific Markovian channels： amplitude damping channel, phase damping channel, depolarizing channel and bit flit channel for single-qnbit states. We show that the amplitude damping channel, phase damping channel, and depolarizing channel do not change the coherence-induced state ordering by l1 norm of coherence, relative entropy of coherence, geometric measure of coherence, and Tsallis relative α-entropies, while the bit flit channel does change for some special cases.

Dual mean Minkowski measures of symmetry for convex bodies

We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which is in a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well.

Quantum Correlations in Heisenberg XY Chain

Quantum correlations measured by quantum discord （QD）, measurement-induced distance （MID）, and geometric measure of quantum discord （GMQD） in two-qubit Heisenberg XY spin chain are investigated. The effects of DM interaction and anisotropic on the three correlations are considered. Characteristics of various correlation measures for the two-qubit states are compared. The increasing Dz increases QD, MID and GMQD monotonously while the increasing anisotropy both increases and decreases QD and GMQD. The three quantum correlations are always existent at very high temperature. MID is always larger than QD, but there is no definite ordering between QD and GMQD. PACS numbers： 03.65.Ta, 03.67.Mn