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

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