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We develop a Gaussian state treatment that allows a transparent quantumdescription of the continuous, nondestructive imaging of and feedback on aBose-Einstein condensate. We have previously demonstrated [Phys. Rev. Lett.\textbf{115}, 060401 (2015)] that the measurement back action of stroboscopicimaging leads to selective squeezing and entanglement of quantized densityoscillations. Here, we investigate how the squeezing and entanglement areaffected by the finite spatial resolution and geometry of the probe laser beamand of the detector and how they can be optimized.

Manipulation of collective quantum states in Bose-Einstein condensates by continuous imaging