  We discuss the entanglement spectrum of the ground state of a gapped
(1+1)-dimensional system in a phase near a quantum phase transition. In
particular, in proximity to a quantum phase transition described by a conformal
field theory (CFT), the system is represented by a gapped Lorentz invariant
field theory in the "scaling limit" (correlation length $\xi$ much larger than
microscopic 'lattice' scale $a$), and can be thought of as a CFT perturbed by a
relevant perturbation. We show that for such (1+1) gapped Lorentz invariant
field theories in infinite space, the low-lying entanglement spectrum obtained
by tracing out, say, left half-infinite space, is precisely equal to the
physical spectrum of the unperturbed gapless, i.e. conformal field theory
defined on a finite interval of length $L_\xi=$ $\log(\xi/a)$ with certain
boundary conditions. In particular, the low-lying entanglement spectrum of the
gapped theory is the finite-size spectrum of a boundary conformal field theory,
and is always discrete and universal. Each relevant perturbation, and thus each
gapped phase in proximity to the quantum phase transition, maps into a
particular boundary condition. A similar property has been known to hold for
Baxter's Corner Transfer Matrices in a very special class of fine-tuned, namely
integrable off-critical lattice models, for the entire entanglement spectrum
and independent of the scaling limit. In contrast, our result applies to
completely general gapped Lorentz invariant theories in the scaling limit,
without the requirement of integrability, for the low-lying entanglement
spectrum. - The finite-size spectra of the entanglement Hamiltonian and of the
physical Hamiltonian on an interval of length R exhibit entirely different
behaviors upon crossover from the critical regime $R \ll \xi$ to the gapped
regime $R \gg \xi$.
