distance distance 14 and discuss the lo calization of the non-abelian curva- ture near the fixe d quasiholes. The difference between the Berry phases for odd and even number of electrons, shown in Fig. 8, converges extremely fast to π [the plot suggests an exp onen tial convergence]. From the connec- tion made in the previous paragraph, this implies that the non-abelian curv ature is also strongly [exponentially] localized near the fixed anyons. Unfortunately, the size of our systems is too small to allow a quantitative eval- uation of this asymptotic decay behavior. Fig. 13 tells us how far are we from a converged situation, where the open circles should overlap with the continuous line. We end this discussion Section by mentioning that we did compute sev eral braiding monodromies and tried to compare their group properties under multiplication with the predictions follo wing from the underlying CFT struc- ture of the Mo ore-Read state derived in Ref. 19. Unfor- tunately, the size of our system is too small to see a strong correlation between the two. This can be seen from Fig. 13, where the continuous line practically repre- sent the thermo dynamic limit of the non-abelian curva- ture and the open circles represent the same quantity for our largest finite system. The most problematic part is the large discrepancy seen at points in between the quasi- holes (the end of the curves in Fig. 13), which affects any monodromy looping around a fixed anyon. Bratelli diagram what these sectors are: they should be the q-spin S =0 and S =1 sectors. 13 For the sphere geom- etry, these sectors correspond to the even, respectively the odd numbers of electrons. FIG. 13: (Color online.) Comparison between the results obtained for the Abelian case (one flux added) and for the non-Abelian case (two flux added). The left/right panels refer to odd/even number of electrons. The solid lines in the upper/lower panels is the difference/sum between the Abelian Berry curvature for N/N$_{φ}$ =16/30 and N/N$_{φ}$ =15/29 (see Fig. 9). The two open circle lines in the upper/lower pan- els show the values (multiplied by 2) of f$_{0}$ ( x )/ | f ( x ) | recorded along the path shown in the inset. In the left panels, the dot- ted lines correspond to the system sizes N/N$_{φ}$ =15/29 and 13/25. In the left panels, the dotted lines correspond to N/N$_{φ}$ =14/27 and 12/23. We have shown that the Hall sequences can be viewed as the zero modes of certain Hamiltonians, which were written down explicitly using many-particle creation op- erators. The compressible Hall states with the anyons fixed at definite locations were shown to be the zero modes of a pinning Hamiltonian, which was also written down explicitly. We have developed an efficient diago- nalization algorithm for the pinning potential, which was subsequently used to explore some general properties and to map the quantum geometry of the Moore-Read states with two and four anyons. Working with two anyons, we were able to give the first direct confirmation of the topological properties of the monodromies. The monodromy corresponding to a loop enclosing one of the anyons was found to differ by exactly a factor -1, when computed for even/odd num- ber of electrons (which correspond to the q-spin S =0/1 sectors). This is the first explicit confirmation of the the- oretical predictions based on the underlying CFT struc- ture of the Moore-Read sequence. IX. CONCLUSIONS By fusing the two anyons, we found that the electron density is different for the even/odd number of electrons. This is the first direct confirmation that one can deter- mine if a quantum state is in the S =0/1 sectors by fusing the anyons and measuring the electron density. This type of measurement stands at the basis of read-in and read- out processes of TQC. 13 Working with four anyons, we mapped the abelian and Once we made this connection, we can we can go back To demonstrate that this is the case, we notice that | f ( P ) | plotted in Fig. 10 is equal to half the difference between the two eigenvalues of the curvature coefficient ˆ F ( P ) and that Tr ˆ F ( P ) plotted in Fig. 9 is equal to the sum of the two eigenvalues. Thus, near the fixed quasi- holes, the values of these quantities, properly normalized, should compare well with the values shown in the plots of the lower panels of Fig. 7. To verify that this is the case, we generated line-plots out the surface plots shown in Figs. 10 and 9. The line plots were generated by record- ing the points along the path shown in the inset of Fig. 13. We used the data corresponding to the standard metric of the sphere. These line plots are compared in Fig. 13 with the absolute value of the data shown in Fig. 7. As one can see, the matching is almost perfect near the fixed anyon. By comparing two system sizes, we can also see the agreement becoming better as the size of the system is increased. We expect the agreement to become perfect as the sphere radius is taken to infinity. It is important to notice that there is agreement for both even and odd number of electrons cases.