  In this paper, we study the geometry of reduced density matrices for states
with symmetry-protected topological (SPT) order. We observe ruled surface
structures on the boundary of the convex set of low dimension projections of
the reduced density matrices. In order to signal the SPT order using ruled
surfaces, it is important that we add a symmetry-breaking term to the boundary
of the system---no ruled surface emerges in systems without boundary or when we
add a symmetry-breaking term representing a thermodynamic quantity. Although
the ruled surfaces only appear in the thermodynamic limit where the
ground-state degeneracy is exact, we analyze the precision of our numerical
algorithm and show that a finite system calculation suffices to reveal the
ruled surface structures.
