The entanglement or separability of mixed quantum states as a matter of the choice of observables

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.