Much about a quantum system is captured by the collection of its classical subsystems, such as algebraic, logical, and information-theoretic aspects. However, one can rigorously prove that the collection of classical subsystems cannot capture all information about a quantum system, for which more structure has to be added. This question is answered by the notion of an active lattice, which adds the information of how to switch between two classical viewpoints. After a survey of this area, I will discuss how to characterise active lattices, and how to reconstruct an operator algebra from its active lattice, parallelled with many illustrations from piecewise Boolean algebra.
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