  The bulk electric polarization works as a nonlocal order parameter that
characterizes topological quantum matters. Motivated by a recent paper [H.
Watanabe \textit{et al.}, Phys. Rev. B {\bf 103}, 134430 (2021)], we discuss
magnetic analogs of the bulk polarization in one-dimensional quantum spin
systems, that is, quantized magnetizations on the edges of one-dimensional
quantum spin systems.The edge magnetization shares the topological origin with
the fractional edge state of the topological odd-spin Haldane phases. Despite
this topological origin, the edge magnetization can also appear in
topologically trivial quantum phases. We develop straightforward field
theoretical arguments that explain the characteristic properties of the edge
magnetization. The field theory shows that a U(1) spin-rotation symmetry and a
site-centered or bond-centered inversion symmetry protect the quantization of
the edge magnetization. We proceed to discussions that quantum phases on
nonzero magnetization plateaus can also have the quantized edge magnetization
that deviates from the magnetization density in bulk. We demonstrate that the
quantized edge magnetization distinguishes two quantum phases on a
magnetization plateau separated by a quantum critical point. The edge
magnetization exhibits an abrupt stepwise change from zero to $1/2$ at the
quantum critical point because the quantum phase transition occurs in the
presence of the symmetries protecting the quantization of the edge
magnetization. We also show that the quantized edge magnetization can result
from the spontaneous ferrimagnetic order.
