The Determinability Of The Quantum State By Its Subsystems

In the research of the quantum entanglement theory, the "parts and whole" problem with quantum entangled states is one of the key problems. In the "parts and whole" problem, we mainly study on the correlations between quantum entangled states and their reduced density matrices. We can get some useful entanglement information from the solution of the "parts and whole" problem, this helps us to understand the structure of entanglement. Furthermore, the "parts and whole"problem is also important in condensed-matter physics and chemical physics. In this dissertation, we study the "parts and whole" problem of the entangled states with wide application, and solve the problem of the uniquely determinabilty by their reduced density matrices. In addition to these, we also find the determining sets with them. The details are as follows.For W-type states, we prove that all of the n-qubit W-type states can be uniquely determined (among pure, mixed states) by their (n -1)bipartie reduced density matrices, whose index is a set corresponds to a tree graph. Our result shows the invariant property of the W state under the stochastic local operations and classical communication equivalent,and generalizes the conclusion of the previous, which is "the index of the bipartite reduced density matrices is a set corresponds to a star graph or a line graph can uniquely determine the W-type states"For the determinability of the Dicke-type states, we prove them can be reduced to (l+1)-partite level. We use the property of combination and construct two kinds of the (l +1) -partite reduced density matrices sets, and prove that both of them can uniquely determine the Dicke-type states. More importantly, the number of elements in these two subsets is much smaller than Cn-1l,. And from the construction process, we can get that the two kinds reduced density matrices subsets are more closer to the correlation structure of the Dicke-type states.Next, for the stabilizer states, their irreducible k -party correlation has been got, but their determining sets have not been obtained yet. In this dissertation, we prove that all n-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for subsystems which are the supports of the independent generators of the corresponding stabilizer formalisms. Our result can provide some insight into the characterization of multiparty entanglement with the stabilizer states.