  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 a
certain colouring problem. We consider two problems. Firstly we find the most
general way of manipulating the state to obtain maximally entangled states.
After such a manipulation the entangled state |11>+|22>+....|mm> is obtained
with probability p_m. We obtain an expression for the optimal average
entanglement. Also, some results of Lo and Popescu pertaining to this problem
are given simple geometric proofs. Secondly, we consider how to manipulate one
two particle entangled pure state to another with certainty. We derive
Nielsen's theorem (which states the necessary and sufficient condition for this
to be possible) using the method of areas.

Quantum Physics