Method of areas for manipulating the entanglement properties of one copy of a two-particle pure entangled state

We consider the problem of how to manipulate the entanglement properties of a general two-particle pure state, shared between Alice and Bob, by using only local operations at each end and classical communication between Alice and Bob. A method is developed in which this type of problem is found to be equivalent to a problem involving the cutting and pasting of certain shapes along with certain coloring problems. We consider two problems. First, we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation, the entangled states $|11〉+|22〉+\cdot \cdot \cdot +|\mathrm{mm}〉$ are obtained with probability $p_m.$ We obtain an expression for the optimal average entanglement obtainable. Also, some results of Lo and Popescu (e-print quant-ph/9707038) pertaining to this problem are given simple geometric proofs. Second, we consider how to manipulate one two-particle entangled state $|\psi 〉$ to another $|{\psi }^{\prime }〉$ with certainty. We derive Nielsen’s theorem (which states a necessary and sufficient condition for this to be possible) using the method of areas.