Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer

Multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states, are examined. The geometric measure of entanglement of a qubit with other qubits is found for the graph states represented by arbitrary graphs. The entanglement depends on the degree of the vertex representing the qubit, the absolute values of the parameter of the phase shift gate and the parameter of state the gate is acting on. Also the geometric measure of entanglement of the graph states is quantified on the quantum computer ibmq athens. The results obtained on the quantum device are in good agreement with analytical ones.

The Tightness of Multipartite Coherence from Spectrum Estimation

Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods by using the same experimental data [Ding et al. Phys. Rev. Research 3, 023228 (2021)]. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.

Causal Information Rate

Information processing is common in complex systems, and information geometric theory provides a useful tool to elucidate the characteristics of non-equilibrium processes, such as rare, extreme events, from the perspective of geometry. In particular, their time-evolutions can be viewed by the rate (information rate) at which new information is revealed (a new statistical state is accessed). In this paper, we extend this concept and develop a new information-geometric measure of causality by calculating the effect of one variable on the information rate of the other variable. We apply the proposed causal information rate to the Kramers equation and compare it with the entropy-based causality measure (information flow). Overall, the causal information rate is a sensitive method for identifying causal relations.

Self-intersections of closed parametrized minimal surfaces in generic Riemannian manifolds

This article shows that for generic choice of Riemannian metric on a compact oriented manifold M of dimension four, the tangent planes at any self-intersection $$p \in M$$ p ∈ M of any prime closed parametrized minimal surface in M are not simultaneously complex for any orthogonal complex structure on M at p. This implies via geometric measure theory that $$H_2(M;{{\mathbb {Z}}})$$ H 2 ( M ; Z ) is generated by homology classes that are represented by oriented imbedded minimal surfaces.

On the constancy theorem for anisotropic energies through differential inclusions

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices $$C_f$$ C f that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if f is assumed to be non-negative, then in $$C_f$$ C f there is no $$T'_N$$ T N ′ configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find $$T'_N$$ T N ′ configurations in $$C_f$$ C f , but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.

What is a reduced boundary in general relativity?

The concept of boundary plays an important role in several branches of general relativity, e.g. the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e. the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in–out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.

Modelling Geometric Measure of Variation About the Population Mean

Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measureshave allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean.Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standarddeviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyedthe algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean