  In quantum Darwinism, the pointer observable of a system leaves redundant
imprints in its environment after decoherence. Each imprint is recorded in a
fraction of the environment, which identifies a particular partition of the
environment. An ambiguity situation may occur when another observable
noncommuting to the pointer observable also leaves redundant imprints with
respect to another partition of the environment. We study this problem based on
a uniqueness theorem we proved. We find that within a particular subset of all
possible partitions of the environment, the observable of the system leaving
redundant and nondegenerately recorded imprints in the environment is unique.
And, in a typical situation, the partitions outside this particular subset have
no physical significance.
