  In this paper we study the effect of non-trivial spatial topology on quantum
entanglement by examining the degenerate ground states of a topologically
ordered system on torus. Using the string-net (fixed-point) wave-function, we
propose a general formula of the reduced density matrix when the system is
partitioned into two cylinders. The cylindrical topology of the subsystems
makes a significant difference in regard to entanglement: a global quantum
number for the many-body states comes into play, together with a decomposition
matrix $M$ which describes how topological charges of the ground states
decompose into boundary degrees of freedom. We obtain a general formula for
entanglement entropy and generalize the concept of minimally entangled states
to minimally entangled sectors. Concrete examples are demonstrated with data
from both finite groups and modular tensor categories (i.e., Fibonacci, Ising,
etc.), supported by numerical verification.
