The Separability Criteria Of Quantum States

In this paper, we mainly investigate two kinds of quantum states, which are genuine en-tangled state and fully separable state. Quantum entanglement is an important feature whichcan differentiate quantum mechanics and classical mechanics, and is a kind of new resourcesbeyond the classical resources. It is helpful for tasks which are impossible or difficult for theclassical resources. So the study of quantum entanglement is particularly important, and in re-cent years, it has attracted wide attention. And the corresponding states of entangled states areseparable states, so the researches of separable states are also very important significance.A qualitative analysis of the entanglement and the separability of quantum states is animportant task. In general, there are two methods to determine the quantum state whether isan entangled state. One method is carried by Bell inequalities, and another method is carriedby Optional Witness . However, due to the complexities and limitations of these methods, itsconclusions could hardly satisfy us. In this paper, we will analyze entanglement and separabilityof quantum states from two kinds of the perspective. One is constructing Optimal Witness togive a sufficient condition for given quantum state being a genuine entangled state, and themethod simplifies the complexity of the problem. In addition, from a new point of view, thatis, the angle of the density matrix elements, respectively we also obtain sufficient conditions ofgenuine entanglement states and fully separable states.The prior knowledge and the definitions of genuine entangled state and fully separablestate are given in the first chapter.For a given quantum state how to determine whether is a genuine entangled state, that is,not biseparable, is a very important matter. Constructing Optimal Witness can give a discrimi-nant method. Any a given pure state can find Optimal Witness byW|ψ? =αI?|ψ??ψ|, whereα= |m??a∈xD |??|ψ?|2, where D denotes the set of fully separable state.However, it is not a simple task to get the specific value ofαfor us.Chapter II, first of all, we construct the Optimal Witness WnGHZ for the most commonlyused GHZ state, and according to the characteristics of GHZ state we can find a simple methodof computingα. Secondly, we get the Optimal Witness W|ψg? for the GHZ Class |ψg?in a similarway .Chapter III, by the use of the second chapter of the WnGHZ and W|ψg?, we can get a sufficientcondition to determine a genuine entangled state for a given quantum state. In this paper, we get the conditions satisfied by elements of the coefficients for a given pure state being a genuineentangled state. In addition, we also give the conditions which the density matrix elementsmeet for mixed states being a genuine mixed state entanglement. Finally for the common stateρ= p|φnW??φnW| + ( 1 ? p )|φnGHZ??φnGHZ|, by using WnGHZ and W|ψg?, we determine thatρis a genuine entangled state if p satisfies some given conditions, and we can obtain differentresults for different Optimal Witness, so the given conclusion is a sufficient condition, ratherthan necessary and sufficient condition.In chapter IV, let'us think about the matrix elements. Ifρis biseparable state, then thematrix elements must fulfill some limits, that is, violation implies a genuine entangled state.Firstly, when given state is biseparable state, the limits of density matrix elements must bemet. In addition, multi-body and multi-level'conclusion is also got by similar methods, and itoffers the sufficient condition of genuine entangled states. In addition, in this chapter, we willconcern about the separability of quantum states. From matrix elements of the quantum state,we shows a sufficient condition for many-body two-level quantum states to determine fullyseparable states, which is generalized to multi-body multi-level. It is valuably indicated thatthe sufficient condition is necessary and sufficient condition for a class of common quantumstateρ(p) = ( 1 ? p )|ψnGHZ?ψnGHZ| + p 2In. At the end of this chapter, we will prove it.The research of the separability and the entanglement is necessary, and is necessary fromvarious angles, but so far, it did not achieve good results and much progress, so we must con-tinuously try to work.