Bipartite entanglement in AJL's algorithm for three-strand braids

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.