ar X iv : q ua nt - p h / 01 03 15 8 v 1 2 9 M ar 2 00 1 More Is Not Necessarily Easier

We consider pure bipartite states shared by Alice and Bob. Let A and B be the labels of their respective quantum systems, each with d dimensions. Suppose Alice and Bob want to transform a state |ψ〉 to another state |φ〉, and they each have complete classical descriptions of the states. We focus on exact and deterministic transformation using only Local quantum Operations and Classical Communication (LOCC) [1]. If the transformation is possible, we write |ψ〉 → |φ〉, otherwise, we write |ψ〉 6→ |φ〉. A necessary and sufficient condition for |ψ〉 → |φ〉 was found by Nielsen [2], with beautiful connections to the theory of majorization. For a pure state |η〉, let λη denote the list of eigenvalues of the reduced density matrix TrA(|η〉〈η|) in descending order, and let λ (i) η be the i-th element in λη. Nielsen’s criteria can be stated as