Rank-bmRpositive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors

The Hermitian tensor is regarded as an extension of the Hermitian matrix and can be used to represent a quantum mixed state. In quantum information, the problem of separability discrimination and decomposition of a quantum mixed state is still important and hard. In this paper, we deduce the gradient of the approximation function, and propose three algorithms: A negative gradient algorithm and a BFGS algorithm for rank-$R$ positive Hermitian approximation of Hermitian tensors, and a separability discrimination and decomposition algorithm for Hermitian tensors. According to the Taylor formula and the convexity analysis, we prove the effectiveness of the algorithm. Numerical examples also verify the correctness of the theoretical analysis and the effectiveness of algorithms. They show that the BFGS algorithm can be used for the separability discrimination and the positive Hermitian decomposition, as well as to obtain a rank-positive Hermitian decomposition. Compared with the semidefinite relaxation algorithm, the BFGS algorithm has the advantages of less running time and solving the decomposition of higher-order or higher-dimensional Hermitian tensors.