I’m in search of an authoritative reference giving a foundational/information-theoretic approach to *classical* measurement. What abstract physical properties are necessary and sufficient?

Motivation: The Copenhagen interpretation treats the measurement process as a fundamental primitive, and this persists in most uses of quantum mechanics outside of foundations. Of course, the modern view is that the measurement process is just another physical evolution, where the state of a macroscopic apparatus is conditioned on the state of a microscopic quantum system in some basis determined by their mutual interaction Hamiltonian. The apparent nonunitary aspects of the evolution inferred by the observer arises because the measured system is coupled to the observer himself; the global evolution of the system-apparatus-observer system is formally modeled as unitary (although the philosophical meaningfulness/ontology/reality of the components of the wavefunction corresponding to different measurement outcomes is disputed).

Eventually, we’d like to be able to identify all laboratory measurements as just an anthropocentric subset of wavefunction branching events. I am very interested in finding a mathematically precise criteria for branching.^{a } Ideally, I would like to find a property that everyone agrees must apply, at the least, to laboratory measurement processes, and (with as little change as possible) use this to find all branches — not just ones that result from laboratory measurements.^{b }

It seems sensible to begin with what is necessary for a *classical* measurement since these ought to be analyzable without all the philosophical baggage that plagues discussion of quantum measurement. But as far as I can tell, very little has been written about this topic. When I search for things like “classical measurement theory”, I get a lot of results about error propagation, estimators, etc., which isn’t at all what I’m interested in. Rather, I’d like to better understand how to think about the necessary and sufficient *physical* conditions for measurements and amplification to happen from first principle. What does the initial joint state of the system and apparatus have to be like and (especially) what sorts of interactions are needed?

This is not completely trivial. As I first heard emphasized by Sean Carroll, measurements (whether classical or quantum) rely strongly on negentropy as a resource. Any time you map the state of one system to another with a unitary/reversible process, you must have a “blank tape” (or blank hard drive) on which to write the answer. The vast majority of states in a classical phase space are near-maximal entropy, and therefore unsuitable for making measurement. Furthermore, useful measurements seem closely connected to amplification. Even classically, a sensitive measurement device is one which maps the states of a very small (often microscopic) system to some states of the apparatus. The apparatus invariably has many more possible noisy states, and so the measurement process defines a one-to-many mapping that divides up the state space of the apparatus into equivalence classes all denoting the same measurement outcome. Such a process is predicated on some sort of sensitive dependence on initial condition, and presumably can be connected to the study of chaos.

I think there’s much more to be said along these lines, touching on other aspects like stability and repeatability. Where should I look?

**Edit (2021-Jan-3)**: This recent paper explores the closely-related question of what sorts of theories (classical, quantum, and from the “general probabilistic theory” operational framework) admit something like quantum Darwinism. Comments from me here.

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