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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?

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  • $\begingroup$ Entanglement doesn't change the physical properties of a system. It changes the correlations between multiple measurements on the system. Moreover, it's somewhat fickle in the sense that the more subsystems participate in an entangled state, the less the individual correlations can be, which is captured in "entanglement monogamy". $\endgroup$
    – CuriousOne
    Apr 8, 2016 at 4:33
  • $\begingroup$ @CuriousOne Are you making a distinction between the "physical properties" of a system and the list of the results of every conceivable set of measurements one might perform on that system? And, if so, what is that distinction? $\endgroup$
    – Rococo
    Apr 8, 2016 at 5:05
  • $\begingroup$ @Rococo: By that I mean that entangled photon states don't change their energy or pick up an effective mass as they would by an actual physical interaction. The individual measurement does not change in a noticeable way, which is different from what an actual physical interaction would do. I may be wrong but I don't think this will be different for the case of a BEC. $\endgroup$
    – CuriousOne
    Apr 8, 2016 at 5:09
  • $\begingroup$ Does a BEC of entangled particles still retain superfluidity (no friction) due to identical movement direction? $\endgroup$ Apr 8, 2016 at 5:10
  • $\begingroup$ @QuantumJournalist- under some circumstances this will be true. Superfluid helium is a strongly interacting system, in which the true many-body state is presumably some complicated entangled state, but it is still the prototypical example of a superfluid. $\endgroup$
    – Rococo
    Apr 8, 2016 at 20:19

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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...

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  • $\begingroup$ Your two ways of defining a BEC are in fact identical (i.e. one implies the other). $\endgroup$
    – Adam
    Apr 8, 2016 at 9:41
  • $\begingroup$ @Adam- not exactly. For example, a system that has macroscopic values of several terms in the single-particle density matrix (a 'fragmented BEC') does not have ODLRO, and so is a BEC by one definition and not the other. $\endgroup$
    – Rococo
    Apr 8, 2016 at 20:01

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