On The Local Invariants Of Pure Four-qubit States

By using the methods of group theory, we study the local invariants of pure four-qubit states and the locally unitary transformations. Two n-qubit states are equivalent or they have the same entanglement properties, if they are on the same orbit of the local unitary group, U(2)(?)n. On the one hand, we get the number of parameters needed to describe inequivalent four-qubit states, namely the invariants’dimension of four-qubit states under local unitary transformations; On the other hand, we derive the invariants of degree of≤6in detail under local unitary transformations, but show there is no complete set of algebraically independent polynomial invariants of degree no more than6. At last, we also list the value of some invariants of some special four-qubit states.In chapter Ⅰ, we mainly introduces some basic concepts used in the article.In chapter Ⅱ, the number of parameters needed to describe the non-local properties of inequivalent pure four-qubit states are discussed. Simultaneously, We show the number for describing a pure four-qubit state can be reduced from thirty-two to twenty-two by local unitary transformations.In chapter Ⅲ, we give the local invariants of pure four-qubit state whose degrees are no more than six. And we show that the dimension of invariants degree≤6is no more than seventeen.In chapter Ⅳ, we exhibit the invariants of some special states such as GHZ state.