  The reconstruction of the state of a multipartite quantum mechanical system
represents a fundamental task in quantum information science. At its most
basic, it concerns a state of a bipartite quantum system whose subsystems are
subjected to local operations. We compare two different methods for obtaining
the original state from the state resulting from the action of these
operations. The first method involves quantum operations called Petz recovery
maps, acting locally on the two subsystems. The second method is called matrix
(or state) reconstruction and involves local, linear maps which are not
necessarily completely positive. Moreover, we compare the quantities on which
the maps employed in the two methods depend. We show that any state which
admits Petz recovery also admits state reconstruction. However, the latter is
successful for a strictly larger set of states. We also compare these methods
in the context of a finite spin chain. Here, the state of a finite spin chain
is reconstructed from the reduced states of a few neighbouring spins. In this
setting, state reconstruction is the same as the MPO (i.e. matrix product
operator) reconstruction proposed by Baumgratz et al. in [Phys. Rev. Lett. 111,
020401 (2013)]. Finally, we generalize both these methods so that they employ
long-range measurements instead of relying solely on short-range correlations
embodied in such local reduced states. Long-range measurements enable the
reconstruction of states which cannot be reconstructed from measurements of
local few-body observables alone and hereby we improve existing methods for
quantum state tomography of quantum many-body systems.
