  In a quantum measurement process, classical information about the measured
system spreads throughout the environment. Meanwhile, quantum information about
the system becomes inaccessible to local observers. Here we prove a result
about quantum channels indicating that an aspect of this phenomenon is
completely general. We show that for any evolution of the system and
environment, for everywhere in the environment excluding an $O(1)$-sized region
we call the "quantum Markov blanket," any locally accessible information about
the system must be approximately classical, i.e. obtainable from some fixed
measurement. The result strengthens the earlier result of arXiv:1310.8640 in
which the excluded region was allowed to grow with total environment size. It
may also be seen as a new consequence of the principles of no-cloning or
monogamy of entanglement. Our proof offers a constructive optimization
procedure for determining the "quantum Markov blanket" region, as well as the
effective measurement induced by the evolution. Alternatively, under
channel-state duality, our result characterizes the marginals of multipartite
states.
