  Motivated by the idea of entanglement loss along Renormalization Group flows,
analytical majorization relations are proven for the ground state of
(1+1)-dimensional conformal field theories. For any of these theories,
majorization is proven to hold in the spectrum of the reduced density matrices
in a bipartite system when changing the size L of one of the subsystems.
Continuous majorization along uniparametric flows is also proven as long as
part of the conformal structure is preserved under the deformation and some
monotonicity conditions hold as well. As particular examples of our
derivations, we study the cases of the XX, Heisenberg and XY quantum spin
chains. Our results provide in a rigorous way explicit proves for all the
majorization conjectures raised by Latorre, Lutken, Rico, Vidal and Kitaev in
previous papers on quantum spin chains.
