Observational Entropy and Quantum Darwinism

Note to Readers: If you haven't spent much time thinking about quantum theory, please skip this one. But if you have, please give me any feedback you might have on X. This was a fun project for me that challenged me to remember things I haven't thought much about since grad school. I appreciated the encouragement from Anthony Aguirre to try to formalize what were intuitions and this is at least the initial result of that effort.

The Informational Structure of Einselection: Observational Entropy in Interaction-Free Measurements

The transition from quantum superpositions to objective classical reality is characterized by the selection of a preferred basis (einselection) and the proliferation of information (Quantum Darwinism). Zurek’s “predictability sieve” identifies pointer states through a dynamical criterion: states that survive are those the environment fails to scramble. We demonstrate that this dynamical selection corresponds to an information-theoretic criterion: the minimization of Observational Entropy (OE). While decoherence provides the mechanism for basis selection, OE provides the accounting. The pointer basis emerges as the unique coarse-graining that balances the observer’s information gain against the system’s coherence loss—a trade-off we term the informational saddle point. We test this correspondence by analyzing the High-Efficiency Quantum Interrogation experiment of Kwiat, Zeilinger, et al. (1999). Their Figure 3 reveals an efficiency peak at finite N, where N is the number of interrogation cycles. We show that this peak coincides with the minimum of Observational Entropy, providing empirical evidence that dynamical stability (the sieve) and informational efficiency (OE minimization) identify the same classical limit.