  We propose an order parameter for the Symmetry-Protected Topological (SPT)
phases which are protected by Abelian on-site symmetries. This order parameter,
called the "SPT-entanglement", is defined as the entanglement between A and B,
two distant regions of the system, given that the total charge (associated with
the symmetry) in a third region C is measured and known, where C is a connected
region surrounded by A, B and the boundaries of the system. In the case of
1-dimensional systems we prove that in the limit where A and B are large and
far from each other compared to the correlation length, the SPT-entanglement
remains constant throughout a SPT phase, and furthermore, it is zero for the
trivial phase while it is nonzero for all the non-trivial phases. Moreover, we
show that the SPT-entanglement is invariant under the low-depth quantum
circuits which respect the symmetry, and hence it remains constant throughout a
SPT phase in the higher dimensions as well. Also, we show that there is an
intriguing connection between SPT-entanglement and the Fourier transform of the
string order parameters, which are the traditional tool for detecting SPT
phases. This leads to a new algorithm for extracting the relevant information
about the SPT phase of the system from the string order parameters. Finally, we
discuss implications of our results in the context of measurement-based quantum
computation.
