  We propose to utilize the sub-system fidelity (SSF), defined by comparing a
pair of reduced density matrices derived from the degenerate ground states, to
identify and/or characterize symmetry protected topological (SPT) states in
one-dimensional interacting many-body systems. The SSF tells whether two states
are locally indistinguishable (LI) by measurements within a given sub-system.
Starting from two polar states (states that could be distinguished on either
edge), the other combinations of these states can be mapped onto a Bloch
sphere. We prove that a pair of orthogonal states on the equator of the Bloch
sphere are LI, independently of whether they are SPT states or cat states
(symmetry-preserving states by linear combinations of states that break
discrete symmetries). Armed with this theorem, we provide a scheme to construct
zero-energy exitations that swap the LI states. We show that the zero mode can
be located anywhere for cat states, but is localized near the edge for SPT
states. We also show that the SPT states are LI in a finite fraction of the
bulk (excluding the two edges), whereas the symmetry-breaking states are
distinguishable. This can be used to pinpoint the transition from SPT states to
the symmetry-breaking states.
