The SU(R) Unitary Group And The Separability Criteria Of Quantum State

With the growing demand for information, people are constantly promoting the Infor-mation Science and Technology. Quantum Information Science and Technology refers to a subject combined by quantum mechanics, information science and computer science, mainly including quantum teleportation and quantum calculation. In quantum information science and technology, the research on quantum separability is one of the important contents of the quantum mechanics. Besides, the study of the unitary group SU(R) and some special quantum states is an important starting point of quantum state research.This paper firstly studies the properties of the special unitary groups SU(R) with the determinant for1. From construction of the typical generators of unitary group SU(R), this paper proves that the typical generators should satisfy an important property. Using the property, this paper also gives the representation of the single-particle quantum state and the relationship of the coefficients.Using the typical generators of the special unitary groups SU(2), this paper studies the properties of symmetric quantum states. Symmetric quantum states are a class of quantum states with special form and sophisticate properties. Starting with the specific form of the typical generators in special unitary group SU(2), this paper shows the specific representations of the density matrix of symmetric quantum state and the correlation these coefficients satisfy. Using this representation, this paper also proves that the PPT criterion is equivalent to the rearrangement criterion when analyzing the symmetric quantum separability. In addition, this paper obtains the relationship between the corresponding correlation coefficients on the||Γρ||criterion.This paper finally analyzes the influence of the trace norm and Frobenius norm to the separability criterion. The trace norm and Frobenius norm are two kind of commonly used norms. Based on the two norms respectively, this paper analyzes the separability of a2x2quantum states, expresses the separable conditions of coefficients respectively under the||Γρ||criterion and the rearrangement criterion. Then this paper gains the conclusions:the judgment of the||Γρ||criterion is stronger than the rearrangement criterion based on the same norm; the influence of the trace norm is stronger than Frobenius norm under the same separability criterion.