  In a recent Letter [Phys. Rev. Lett. 122, 010403 (2019)], an equivalence is
proposed between the so-called Spectrum Broadcast Structure for a
system-multienvironment quantum state, and the conjunction of two
information-theory notions: (a) Strong Quantum Darwinism; and (b) Strong
Independence. Here, we show that the mathematical formulation of condition (b)
by the authors (namely, the pairwise independence of the fragments of the
environment, conditioned on the system), is necessary but not sufficient to
ensure the equivalence. We propose a simple counter-example, together with a
strengthened formulation of condition (b), ensuring the equivalence proposed by
the authors.
