Study Of Bipartite Entanglement In AJL's Algorithm For Three-strand Braids

Recently a celebrated result in quantum computation is the discovery of some quantum algorithms able to solve problems faster than any known classical algorithm,such as Shor's algorithm and Grover's algorithm.But the essence of this acceleration is not clear.Aharonov,Jones and Landau have presented a polynomial quantum algorithm for approximating the Jones polynomial,and we call it AJL's algorithm.The AJL's algorithm is very different from Shor's algorithm as that the AJL's algorithm does not use the Fourier transform.In addition,another reason that we study the AJL's algorithm is that the entanglement in AJL's algorithm is not yet fully understood.When the number of braids is three,we can only use trace closure,so we study the AJL's algorithm based on trace closure.We re-discribe AJL's algorithm for threestrand braids,and the AJL's algorithm is functionally equivalent to the previous AJL's algorithm.However,there are slightly difference between these two algorithms.This difference is that AJL's algorithm involves three pure qubits coupled to a single control qubit,whereas the re-discribed AJL's algorithm involves three work qubits in some mixed state coupled to a single control qubit.In the process of calculation,we use the Peres-Horodecki entanglement criterion to study the entanglement features for all bipartitions.There are four different bipartitions in AJL's algorithm based on three-strand braids.The result shows that there is no entanglement between the control qubit and the work ones.If there is bipartition between the first work qubit and the others,the state is a product state.In other words,the state is a separable state.In addition,We prove a sufficient and necessary condition for its entanglement between the second(third)work qubit and the others.Finally,we study and analyze the relationship between its bipartite entanglement and elementary crossings in the three-strand braid group.We also find that braids whose trace closures are topologically identical might have different entanglement properties in AJL's algorithm.