  We address quantum characterization of anisotropic spin chains in the
presence of antisymmetric exchange, and investigate whether the Hamiltonian
parameters of the chain may be estimated with precision approaching the
ultimate limit imposed by quantum mechanics. At variance with previous
approaches, we focus on the information that may be extracted by measuring only
two neighbouring spins rather than a global observable on the entire chain. We
evaluate the Fisher information (FI) of a two-spin magnetization measure, and
the corresponding quantum Fisher information (QFI), for all the relevant
parameters, i.e. the spin coupling, the anisotropy, and the Dzyaloshinskii
Moriya (DM) parameter. Our results show that the reduced system made of two
neighbouring spins may be indeed exploited as a probe to characterize global
properties of the entire system. In particular, we find that the ratio between
the FI and the QFI is close to unit for a large range of the coupling values.
The DM coupling is beneficial for coupling estimation, since it leads to the
presence of additional bumps and peaks in the FI and QFI, which are not present
in a model that neglects exchange interaction and may be exploited to increase
the robustness of the overall estimation procedure. Finally, we address the
multiparameter estimation problem, and show that the model is compatible but
sloppy, i.e. both the Uhlmann curvature and the determinant of the QFI matrix
vanish. Physically, this means that the state of the system actually depends
only on a reduced numbers of combinations of parameters, and not on all of them
separately.
