Quantum Gaussian networks consist of independent nodes in space and Gaussian quantum operations connecting these nodes, where entangled Gaussian states are shared among the nodes. When quantum operations or measurements are performed at the nodes, the quantum entanglement shared at the nodes propagates throughout the network, forming multipartite entanglement. Characterizing the properties of such multipartite entanglement is an interesting problem in quantum networks, as it not only influences the technological development of quantum networks but also enhances our understanding of multipartite entanglement in network scenarios. We present the functional characteristics of preparable entangled Gaussian states using informatic functions such as Renyi entropy-induced multipartite mutual information and generalizations of squashed entanglement measures. These characteristics reveal that the network structure imposes strong constraints on preparable entangled states, which is distinctly different from standard single multipartite entanglement.

In this talk, we present a generalized scalar auxiliary variable (SAV) method for the time-dependent Ginzburg-Landau equations. The backward Euler method is employed to discretize the temporal derivative of the equations. This method decouples and linearizes the system to avoid solving nonlinear equations at each step. Theoretical analysis demonstrates that the generalized SAV method can preserve the maximum bound principle and energy stability, which is confirmed by numerical results. It also shows that the numerical algorithm is stable.

This report mainly discusses the stability and convergence of two types of second-order variable step-size numerical schemes for the incompressible magnetohydrodynamic equations. Scheme 1 is based on a semi-implicit scheme for handling nonlinear terms and stable lowest-order mixed finite element pairs, resulting in a coupled linear system with variable coefficients. The unconditional stability and optimal error estimates of the numerical solution are established. Scheme 2 is based on explicit handling of nonlinear terms and the ZEC technique, splitting the target problem into a series of single-variable linear equations with constant coefficients. Using the inductive hypothesis method, we provide optimal error estimates of the numerical solution under arbitrarily high-order stable mixed finite element pairs. Numerical examples verify the theoretical analysis results, and the effectiveness of the numerical schemes is validated through an adaptive algorithm.