  Suppose two distant observers Alice and Bob share a pure bipartite quantum
state. By applying local operations and communicating with each other using a
classical channel, Alice and Bob can manipulate it into some other states.
Previous investigations of entanglement manipulations have been largely limited
to a small number of strategies and their average outcomes. Here we consider a
general entanglement manipulation strategy and go beyond the average property.
For a pure entangled state shared between two separated persons Alice and Bob,
we show that the mathematical interchange symmetry of the Schmidt decomposition
can be promoted into a physical symmetry between the actions of Alice and Bob.
Consequently, the most general (multi-step two-way-communications) strategy of
entanglement manipulation of a pure state is, in fact, equivalent to a strategy
involving only a single (generalized) measurement by Alice followed by one-way
communications of its result to Bob. We also prove that strategies with one-way
communications are generally more powerful than those without communications.
In summary, one-way communications is necessary and sufficient for the
entanglement manipulations of a pure bipartite state. The supremum probability
of obtaining a maximally entangled state (of any dimension) from an arbitrary
state is determined and a strategy for achieving this probability is
constructed explicitly. One important question is whether collective
manipulations in quantum mechanics can greatly enhance the probability of large
deviations from the average behavior. We answer this question in the negative
for a specific problem.
