Statistical computation of Boltzmann entropy and estimation of the optimal probability density function from statistical sample

In this work, we investigate the statistical computation of the Boltzmann entropy of statistical samples. For this purpose, we use both histogram and kernel function to estimate the probability density function of statistical samples. We find that, due to coarse-graining, the entropy is a monotonic increasing function of the bin width for histogram or bandwidth for kernel estimation, which seems to be difficult to select an optimal bin width/bandwidth for computing the entropy. Fortunately, we notice that there exists a minimum of the first derivative of entropy for both histogram and kernel estimation, and this minimum point of the first derivative asymptotically points to the optimal bin width or bandwidth. We have verified these findings by large amounts of numerical experiments. Hence, we suggest that the minimum of the first derivative of entropy be used as a selector for the optimal bin width or bandwidth of density estimation. Moreover, the optimal bandwidth selected by the minimum of the first derivative of entropy is purely data-based, independent of the unknown underlying probability density distribution, which is obviously superior to the existing estimators. Our results are not restricted to one-dimensional, but can also be extended to multivariate cases. It should be emphasized, however, that we do not provide a robust mathematical proof of these findings, and we leave these issues with those who are interested in them.Comment: 8 pages, 6 figures, MNRAS, in the pres

Local implementation of nonlocal operations of block forms

We investigate the local implementation of nonlocal operations with the block matrix form, and propose a protocol for any diagonal or offdiagonal block operation. This method can be directly generalized to the two-party multiqubit case and the multiparty case. Especially, in the multiparty cases, any diagonal block operation can be locally implemented using the same resources as the multiparty control-U operation discussed in Ref. [Eisert et al., Phys. Rev. A 62, 052317(2000)]. Although in the bipartite case, this kind of operations can be transformed to control-U operation using local operations, these transformations are impossible in the multiparty cases. We also compare the local implementation of nonlocal block operations with the remote implementation of local operations, and point out a relation between them.Comment: 7 pages, 3 figure

Nucleus-nucleus potential from identical-particle interference

Based on the quantum interference between two-identical-nucleus scattering at energies around the Coulomb barrier, the barrier positions for $^{58}$Ni+$^{58}$Ni and $^{16}$O+$^{16}$O are extracted from Mott oscillations in the angular distributions around 90$^{\circ}$ for the first time. The angle separation of pairs of Mott scattering valleys around 90$^{\circ}$ has a direct relationship with the closest distance between two nuclei in elastic scattering. Together with the barrier height from fusion excitation function, the extracted barrier position provides a sensitive probe to constrain the model predictions for the nucleus-nucleus potential barrier.Comment: 4 figures, accepted for publication in Phys. Lett.