Hello, I'm enjoying the conversation so far, If I might, I wonder if the “single-message” framing is missing one subtlety. It seems that the decisive factor is not just the count of adjustable properties, but the effective rank of independent modes once coupling, noise, and feedback are taken into account.
Formally, you could think of a system’s information capacity as
𝐶eff ∼ rank(𝑀)⋅log2(1+SNR),
where rank (𝑀) is the number of controllable orthogonal modes that survive environmental interaction, and SNR sets the number of usable bits per mode.
Quantum behavior would then arise when the effective rank collapses to one under physical constraints, even if the system has more adjustable parameters in principle. That collapse produces the familiar features:
Complementarity → mode capacity allocated to one observable suppresses orthogonality in the conjugate.
Noncommutativity → sequential probes alter which effective mode survives, because the first probe changes the rank structure.
No-cloning → any attempt to amplify the single surviving mode injects entropy that destroys coherence.
From this perspective, the Heisenberg uncertainty principle captures only the time–bandwidth side (number of independent samples), while Shannon’s formula reminds us that SNR is equally fundamental. Quantum “strangeness” then looks like the limit of small rank (𝑀) and low SNR simultaneously.
A technical question this raises: if one could experimentally engineer environments that keep rank
rank(M)=1 but tune SNR independently (say, by adding controlled spectral noise), would the Born probabilities still hold exactly, or would deviations appear once the information rate falls below a critical threshold?
