  We investigate the nature of spontaneous symmetry breaking in complex quantum
systems by conjecturing that the maximally symmetry breaking quantum ground
states are the most classical ones corresponding to an ordered phase. We make
this argument quantitatively precise by showing that the ground states which
realize the maximum breaking of the Hamiltonian symmetries are the only ones
that: I) are always locally convertible, i.e. can be obtained from all other
ground states by local operations and classical communication, while the
reverse is never possible; II) minimize the monogamy inequality for bipartite
entanglement; III) minimize quantum correlations, as measured by the quantum
discord, for all pairs of dynamical variables and are the only ground states
for which the pairwise quantum correlations vanish asymptotically with the
intra-pair distance.
