  Aharonov, Jones, and Landau [Algorithmica 55, 395 (2009)] have presented a
polynomial quantum algorithm for approximating the Jones polynomial. We
investigate the bipartite entanglement properties in AJL's algorithm for
three-strand braids. We re-describe AJL's algorithm as an equivalent algorithm
which involves three work qubits in some mixed state coupled to a single
control qubit. Furthermore, we use the Peres entanglement criterion to study
the entanglement features of the state before measurements present in the
re-described algorithm for all bipartitions. We show that the state is a
product state relative to the bipartition between the first work qubit and the
others. And it has no entanglement between the control qubit and work ones. We
also prove a sufficient and necessary condition for its entanglement between
the second (third) work qubit and the others. Moreover, we discuss the relation
between its bipartite entanglement and elementary crossings in the three-strand
braid group. We find that braids whose trace closures are topologically
identical may have different entanglement properties in AJL's algorithm.
