The Research Of Full Separability For Four Qubit W-GHZ Mixed States In Noise Environment

Quantum information science is a new interdisciplinary subject which combines quantum mechanics and classical information science.It is one of the most promising subjects in recent years.Because of its great practical application value,it has attracted a lot of attention and research.As a very important physical resource in quantum information theory,quantum entanglement exists in the vast majority of quantum information applications,such as quantum superdense coding,quantum key distribution,quantum teleportation and so on.Quantum entanglement has naturally become one of the core problems in the study of quantum information science,most of which focus on solving the problem of multi-body quantum entanglement.Because of the multipartite quantum entanglement has the characteristics of multi particle number,it will face different entanglement characteristics in the research process,namely full separable state,partially separable state and entangled state.At present,the entanglement properties of two commonly used quantum states that are the four qubit W state and the GHZ state,have been studied preliminarily,but the entanglement properties of the four qubit W-GHZ mixed states have not been studied theoretically.In this paper,we will study the full separability of four qubit W-GHZ mixed states in noisy environment.After comparing the advantages and disadvantages of some existing entanglement criteria,we decide to choose the method of entanglement witness to study the full separability of the quantum states.In the first part of this paper main research content,we will explain the process of proving the necessity of full separability in detail.First,we will introduce the quantum states to briefly.Then,we deduce the relevant calculation formula of the necessity of full separability from the two steps of optimal entanglement witness and matching entanglement witness,and use MATLAB mathematical tools to calculate the necessity results.Finally,the results are sorted out and analyzed,and the strict expression of the entanglement witness near the GHZ and W states is constructed,and stringently prove the entanglement witness near the GHZ state.In the second part,we study the sufficient conditions of full separability for the quantum states in the middle part(GHZ and W states account for a large proportion respectively)on the basis of the necessary conditions of separability.In this part,we will find out the corresponding optimal product states and use them to establish the relevant equations.By solving the equations,we will get the sufficient results of the middle part of the quantum states.