If all the particles of a Bose-Einstein condensate become entangled with each other,does the system still remain a Bose-Einstein condensate? I know that an entangled system is found in a single entangled state and that when you try to observe the individual state of a particle from an entangled system using a reduced density matrix, you observe a mixed state, even though if you observe the whole entangled system, you observe a pure state.
So does a BEC still retains its properties in the presence of entanglement between all of its particles?
 A: An ensemble of interacting particles will, over time, develop entanglement between widely separated parts*, so this is similar to asking whether an interacting system can still be a BEC.
The short answer is yes, but a subtlety is that various authors define BEC in slightly different ways.
One way of defining BEC, as I mention in a recent answer, is the property called "off-diagonal long range order" (ODLRO). This is, roughly, a way of quantifying the idea that a BEC has macroscopic coherence. It doesn't pose any problems to test for this property in a complicated entangled many-body state, and it turns out that in some cases (like weakly repulsively interacting bosons) one does still gets it like in the non-interacting case. See also this answer.
Another way of defining BEC is that it is macroscopic occupancy of a single-particle density matrix. In other words, the reduced density matrix of a single particle, when diagonalized, has (at least) one term that is of order N. This is the sensible generalization of the idea that the BEC is macroscopic occupation of a single-particle wave function. The use of density matrices in the definition allows it to handle cases with entanglement, so again the answer in that an entangled Bose system may be in a BEC phase.
*Unless there is something like many-body localization going on, but that's a subject for another day...
