  Microscopically understanding and classifying phases of matter is at the
heart of strongly-correlated quantum physics. With quantum simulations, genuine
projective measurements (snapshots) of the many-body state can be taken, which
include the full information of correlations in the system. The rise of deep
neural networks has made it possible to routinely solve abstract processing and
classification tasks of large datasets, which can act as a guiding hand for
quantum data analysis. However, though proven to be successful in
differentiating between different phases of matter, conventional neural
networks mostly lack interpretability on a physical footing. Here, we combine
confusion learning with correlation convolutional neural networks, which yields
fully interpretable phase detection in terms of correlation functions. In
particular, we study thermodynamic properties of the 2D Heisenberg model,
whereby the trained network is shown to pick up qualitative changes in the
snapshots above and below a characteristic temperature where magnetic
correlations become significantly long-range. We identify the full counting
statistics of nearest neighbor spin correlations as the most important quantity
for the decision process of the neural network, which go beyond averages of
local observables. With access to the fluctuations of second-order correlations
-- which indirectly include contributions from higher order, long-range
correlations -- the network is able to detect changes of the specific heat and
spin susceptibility, the latter being in analogy to magnetic properties of the
pseudogap phase in high-temperature superconductors. By combining the confusion
learning scheme with transformer neural networks, our work opens new directions
in interpretable quantum image processing being sensible to long-range order.
