Extending echo state property for quantum reservoir computing

The paper’s Definition II.2 introduces NS-ESP as a conditional statement: for each input and pair of initial states there exists a window w such that, if the windowed input variance has positive lim inf, then a normalized state-difference ratio vanishes (their Eq. (2)). The paper then asserts, without additional hypotheses, that “if non-stationary ESP holds, then ESP holds” (following Eq. (2)). Under the literal definition, this implication is false: for inputs whose windowed variance eventually vanishes, the antecedent of the NS-ESP conditional is false, so NS-ESP is vacuously satisfied even when state differences do not decay; hence ESP need not hold. The candidate solution correctly flags this and supplies a finite-state counterexample, while showing how the implication can be salvaged under a natural persistently non-stationary input assumption tied to the NS-ESP-chosen window. By contrast, the paper claims the implication unconditionally. On the other two inclusions, the paper states NS-ESP ⊊ Subset NS-ESP ⊊ Subspace NS-ESP and the candidate proves the non-strict inclusions (with a slightly over-engineered argument for subset⇒subspace). Therefore: paper wrong (re: NS-ESP⇒ESP), model correct. Citations: definition and unconditional claim in the paper’s main text (Eq. (2) and “it follows that…” passages) ; subset/subspace inclusions as stated by the paper .