  We characterize boundedness and compactness of pullback operators under
holomorphic maps between Bargmann spaces of entire holomorphic functions with
quadratic strictly plurisubharmonic exponential weights, extending a result of
Carswell-MacCluer-Schuster obtained in the case of the radial quadratic weight.
We also show that the pullback operator between Bargmann spaces is compact
precisely when it is of trace class, with sub-exponentially decaying singular
values.
