  We study the entanglement spectra of many particle systems in states which
are closely related to products of Slater determinants or products of
permanents, or combinations of the two. Such states notably include the
Laughlin and Jain composite fermion states which describe most of the observed
conductance plateaus of the fractional quantum Hall effect. We identify a set
of 'Entanglement Wave Functions' (EWF), for subsets of the particles, which
completely describe the entanglement spectra of such product wave functions,
both in real space and in particle space. A subset of the EWF for the Laughlin
and Jain states can be recognized as Composite Fermion states. These states
provide an exact description of the low angular momentum sectors of the real
space entanglement spectrum (RSES) of these trial wave functions and a physical
explanation of the branches of excitations observed in the RSES of the Jain
states.
