Genuine Etanglement Of Multipartite Quantum States

This paper mainly studies the detection of genuine entanglement for multipartite quantum states.Firstly,we study the genuine entanglement for tripartite qubit quantum states.By reconstructing density matrix based on partial transposition and the norm of correlation tensors,we obtain the necessary conditions for separability under arbitrary bipartition of tripartite quantum states.In the tripartite qubit quantum systems,we obtain an analytical sufficient condition to detect genuine entanglement of quantum states.We can detect more genuine tripartite entangled states by some detailed examples.Secondly,by using the Bloch representation of density matrices,we study the genuine entanglement in tripartite high dimension quantum systems.In the tripartite quantum systems,we define density matrix according to special unitary group and give the norm of the correlation tensors.Some tensors under different bipartitions is presented and the matrices related to the correlation tensors are constructed.By using the trace norm of the matrix,we present the necessary conditions for separability under arbitrary bipartition of high dimension tripartite quantum states.Moreover,we present an analytical sufficient condition for detect genuine tripartite entanglement.Through detailed examples show that we can detect more genuine entanglement of tripartite qudit quantum states.Finally,we consider the partial transposition and Schmidt decomposition of the density matrix.We study the detection of genuine multipartite entanglement and a complete classification of four-partite quantum states.Necessary conditions have been presented for different separable quantum states in four-partite quantum systems based on Schmidt-norm and tensor product.Then the sufficient conditions for detecting the genuine entanglement of four-partite qubit quantum states and a complete classification are studied.The generalized Schmidt decomposition and partial transposition are used to study the high-dimensional four-partite quantum system.So the sufficient conditions for detecting the genuine entanglement of four-partite qudit quantum states and a complete classification are studied.