  We introduce an entropic quantity for two-dimensional (2D) quantum spin
systems to characterize gapped quantum phases modeled by local commuting
projector code Hamiltonians. The definition is based on a recently introduced
specific operator algebra defined on an annular region, which encodes the
superselection sectors of the model. The quantity is calculable from local
properties, and it is invariant under any constant-depth local quantum circuit,
and thus an indicator of gapped quantum spin-liquids. We explicitly calculate
the quantity for Kitaev's quantum double models, and show that the value is
exactly same as the topological entanglement entropy of the models. Our method
circumvents some of the problems around extracting the TEE, allowing us to
prove invariance under constant-depth quantum circuits.
