  We determine the behaviour of entanglement between regions of space in a Bose
gas of fixed particle number around the critical temperature condensation.
Long-range correlations develop in the Bose-Einstein condensate (BEC) phase
transition and the aim here is to find out whether spatial coherence alone
implies entanglement. We use a purity measure of entanglement to derive an
entanglement witness that detects entanglement between two regions of space in
the BEC. It is shown that spatial coherence between the two regions is
necessary for entanglement with coherence and entanglement becoming equivalent
only when the regions occupy the entire confining volume of the gas. The
probabilities for Bosons to occupy the regions is the only other parameter that
influences the amount of entanglement. We calculate explicitly the amount of
entanglement between two regions for a cigar-shaped harmonic potential and find
that it increases with decreasing temperature. A second entanglement witness is
derived for entanglement between three regions of space, where again spatial
coherence between each pair of regions is precursor to entanglement. It is
shown that the state with the maximum amount of entanglement occurs when the
probabilities for the Bosons to occupy the three regions are equal.

Quantum Physics