  Li and Haldane conjectured and numerically substantiated that the
entanglement spectrum of the reduced density matrix of ground-states of
time-reversal breaking topological phases (fractional quantum Hall states)
contains information about the counting of their edge modes when the
ground-state is cut in two spatially distinct regions and one of the regions is
traced out. We analytically substantiate this conjecture for a series of FQH
states defined as unique zero modes of pseudopotential Hamiltonians by finding
a one to one map between the thermodynamic limit counting of two different
entanglement spectra: the particle entanglement spectrum, whose counting of
eigenvalues for each good quantum number is identical (up to accidental
degeneracies) to the counting of bulk quasiholes, and the orbital entanglement
spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is
related to bulk quasihole physics and the orbital entanglement spectrum is
related to edge physics, our map can be thought of as a mathematically sound
microscopic description of bulk-edge correspondence in entanglement spectra. By
using a set of clustering operators which have their origin in conformal field
theory (CFT) operator expansions, we show that the counting of the orbital
entanglement spectrum eigenvalues in the thermodynamic limit must be identical
to the counting of quasiholes in the bulk. The latter equals the counting of
edge modes at a hard-wall boundary placed on the sample. Moreover, we show this
to be true even for CFT states which are likely bulk gapless, such as the
Gaffnian wavefunction.
