  The fragile nature of quantum information makes it practically impossible to
completely isolate a quantum state from noise under quantum channel
transmissions. Quantum networks are complex systems formed by the
interconnection of quantum processing devices through quantum channels. In this
context, characterizing how channels introduce noise in transmitted quantum
states is of paramount importance. Precise descriptions of the error
distributions introduced by non-unitary quantum channels can inform quantum
error correction protocols to tailor operations for the particular error model.
In addition, characterizing such errors by monitoring the network with
end-to-end measurements enables end-nodes to infer the status of network links.
In this work, we address the end-to-end characterization of quantum channels in
a quantum network by introducing the problem of Quantum Network Tomography. The
solution for this problem is an estimator for the probabilities that define a
Kraus decomposition for all quantum channels in the network, using measurements
performed exclusively in the end-nodes. We study this problem in detail for the
case of arbitrary star quantum networks with quantum channels described by a
single Pauli operator, like bit-flip quantum channels. We provide solutions for
such networks with polynomial sample complexity. Our solutions provide evidence
that pre-shared entanglement brings advantages for estimation in terms of the
identifiability of parameters.
