Uniqueness of the observable leaving redundant imprints in the environment in the context of quantum Darwinism

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.