Optimality and Complexity in Measured Quantum-State Stochastic Processes

The paper proves that a measured cCQS reduces to a finite-state HMC (Prop. 1) and that nonunifilarity is generically induced by measurement (Prop. 3) with clear structural conditions and examples. However, its central claim (Prop. 4) that, generically, the measured process has positive entropy rate and positive statistical complexity dimension (hence uncountably many predictive states) is only sketched and explicitly appeals to a conjectural step about irreducible nonunifilarity implying uncountable causal states; the authors acknowledge related gaps and open problems. Thus, the paper’s argument for Prop. 4 is incomplete. The model’s solution fills in parts of the route using known HMP results (analyticity of entropy rate; filter ergodicity; contraction in Hilbert metric) and an IFS viewpoint, but leans on strong “positivity” and “generic nonoverlap” assumptions and cites results that do not fully close the place-dependent IFS/dimension step in this setting. Hence, it is also incomplete, though directionally sound. Citations in the paper substantiate the setup (Single-State Constant-Measurement protocol; reduction to HMC) and generic nonunifilarity, but the generic complexity claim rests on prior work and a conjecture rather than a complete proof . The memoryless exception (Proposition 5) is correctly identified as nongeneric , and the paper repeatedly emphasizes that uncountably many mixed states and infinite predictive memory are typical but not fully settled as a theorem in this framework .