Research On Some Problems In Quantum Information

This paper is mainly divided into two parts-quantum majorization and quantum entanglement.Quantum majorization is a synthesis subject that applicating majorization theory to quantum systems.Let X and Y be two matrices in Mn×m.One says that X is matrix majorized by Y,in notation X<Y(or Y>X),if there exists a row stochastic ma-trix D in Mm such that X=YD.A linear mapping ? from Mn×m into Mn×m is said to be of preserving matrix majorization if ?(X)<<?(Y)whenever X<Y;In this part,we will show that a linear mapping from Mn×m into Mn×m is of local preserving matrix majoriza-tion(Definition 2.2)if and only if ? is of mixed preserving matrix majorization(Definition 2.3).In the part two,we mainly discuss entanglement and separability of quantum states.The so-called entanglement and separability of the quantum state is that for 2-qubit pure state,if it can be written as the form of 2 quantum tensor products,then we call that the pure state is separable,otherwise it is entangled.the main conclusions we have proved as follows:(1)Any 2-qubit separable pure state is transformed into an entangled state by a non-trivial partial swap channel,Furthermore,on the basis of PPT principle,we will give the suffi-cient conditions for the separability of the output quantum state;(2)Giving the necessary and sufficient condition of entanglement and separability about X state and X state is transformed by a trivial partial swap channel;(3)Discuss the entanglement and separability of a special mixed state-Werner state and Werner state is transformed by a trivial partial swap channel.For several special Werner states,provide the necessary and sufficient condition about entanglement and separability of them.And proving the several Werner states are less entangled.