Study On The Separability Problem Based On Bloch Representation Of Density Matrix

Quantum entanglement is the main resource in quantum information processing tasks,which can achieve tasks that cannot be achieved by classical resources,such as quantum communication and quantum computation.The separability problem studies whether a quantum state is entangled or separable.The Bloch representation of the density matrix uses the local Bloch vector and the correlation matrix to describe the density matrix of the multi-partite quantum system,and the entanglement properties mainly depend on the correlation matrix.This thesis is based on the Bloch representation of the density matrix to study the separability of discrete variable systems.The main contributions of the thesis are divided into three parts:1.A sufficient and necessary condition for the separability problem of a kind of Bloch representation is obtained,which shows that the local Bloch vector has an influ-ence on whether the quantum state is entangled or separable.Then this thesis proved a separability criterion related to the local Bloch vector.This criterion is more effective in identifying bound entangled states than the separability criterion that only depends on the correlation matrix.This criterion is equivalent to the optimized computable cross norm or relignment(CCNR)criterion or a important corollary of covariance matrix(CM)criterion,however the proof in this thesis does not need to use the knowledge of relign-ment or CM criterion.2.The necessary conditions for the Bloch representation on the tripartite quantum system to be physical is provided.The proposed separability criterion is extended to the separability problem of the tripartite quantum system,and the necessary conditions for the separability under a specific division and the completely separability of the tripartite state are obtained.These necessary conditions can detect the tripartite entanglement that could not be detected by some previous work.3.The proposed separability criterion is strengthened by combining the separabil-ity criterion with local filtering transformation and symmetric expansion method.Cal-culate the analytical expression of the combination of local filtering transformation and separability criterion when one of the subsystems is a qubit,and analyze how to select the local filtering transformation to strengthen the separability criterion to a greater ex-tent.Combine the proposed necessary condition for the completely separability of the tripartite quantum state with the symmetric extension method,and use the semi-definite programming toolkit to detect more bound entanglement.