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In quantum Darwinism, the pointer observable of a system leaves redundantimprints in its environment after decoherence. Each imprint is recorded in afraction of the environment, which identifies a particular partition of theenvironment. An ambiguity situation may occur when another observablenoncommuting to the pointer observable also leaves redundant imprints withrespect to another partition of the environment. We study this problem based ona uniqueness theorem we proved. We find that within a particular subset of allpossible partitions of the environment, the observable of the system leavingredundant and nondegenerately recorded imprints in the environment is unique.And, in a typical situation, the partitions outside this particular subset haveno physical significance.

Uniqueness of the Observable Leaving Redundant Imprints in the Environment in the Context of Quantum Darwinism