  We discuss the symmetry-protected topological (SPT) orders for bosonic
systems from an information-theoretic viewpoint. We show that with a proper
choice of the onsite basis, the degenerate ground-state space of SPT orders (on
a manifold with boundary) is a quantum error-correcting code with macroscopic
classical distance, hence is stable against any local bit-flip errors. We show
that this error-correcting property of the SPT orders has a natural connection
to that of the symmetry-breaking orders, whose degenerate ground-state space is
a classical error-correcting code with a macroscopic distance, providing a new
angle for the hidden symmetry-breaking properties in SPT orders. We propose new
types of topological entanglement entropy that probe the STP orders hidden in
their symmetric ground states, which also signal the topological phase
transitions protected by symmetry. Combined with the original definition of
topological entanglement entropy that probes the 'intrinsic topological
orders', and the recent proposed one that probes the symmetry-breaking orders,
the set of different types of topological entanglement entropy may hence
distinguish topological orders, SPT orders, and symmetry-breaking orders, which
may be mixed up in a single system.
