  A spin fractionalizes into matter and gauge fermions in Kitaev's spin liquid
on the honeycomb lattice. This follows from a Jordan-Wigner mapping to
fermions, allowing for the construction of minimal entropy ground state
wavefunction on the cylinder. We use this to calculate the entanglement entropy
by choosing several distinct partitionings. First, by partitioning an infinite
cylinder into two, the $-\ln 2$ topological entanglement entropy is
reconfirmed. Second, the reduced density matrix of the gauge sector on the full
cylinder is obtained after tracing out the matter degrees of freedom. This
allows for evaluating the gauge entanglement Hamiltonian, which contains
infinitely long range correlations along the symmetry axis of the cylinder. The
matter-gauge entanglement entropy is $(N_y-1)\ln 2$ with $N_y$ the
circumference of the cylinder. Third, the rules for calculating the gauge
sector entanglement of any partition are determined. Rather small correctly
chosen gauge partitions can still account for the topological entanglement
entropy in spite of long-range correlations in the gauge entanglement
Hamiltonian.
