  In quantum systems, entanglement corresponds to nonclassical correlation of
nonlocal observables. Thus, entanglement (or, to the contrary, separability) of
a given quantum state is not uniquely determined by properties of the state,
but may depend on the choice of the factorization of the algebra of
observables. In the present work, we expose and systematize some recently
reported results about the possibility to represent a single quantum state as
either entangled or separable. We will distinguish in particular the cases of
pure and mixed states. For pure states, it has been shown that observables can
always be constructed such that any state has any amount of entanglement
possible. For mixed states, the situation is more complex and only partial
results are known: while it is always possible to choose a factorization such
that a state appears separable, a general criterion to determine whether a
state can be represented as entangled is not known. These results will be
illustrated by several examples, the phenomenon of quantum teleportation, and
the geometry of the states of two qubits.
