Quantum Change Intervals: Exact Asymptotic Localization with Collective Measurements

Abstract: We study exact-label minimum-error localization of a transient, calibrated pure-state change occupying one nonempty contiguous interval in an otherwise stationary sequence of independent outputs, allowing arbitrary collective measurements. Let $c=|\langle 0|ψ\rangle|$ be fixed as the sequence length grows. For each fixed known interval length $i$, as the number $N$ of admissible translations tends to infinity, the corresponding Toeplitz symbol yields an exact square-root integral for the asymptotic optimal success probability, and the square-root measurement (SRM) attains the same limit. If the known length $i_n$ and $N_n=n-i_n+1$ both diverge, with no restriction on their ratio, the optimal and SRM success probabilities converge to the one-dimensional Toeplitz functional at the effective compound overlap $c<sup>2$, namely $p_1(c<sup>2)$. Under a uniform prior over all nonempty intervals, the unknown-length physical Gram kernel is not globally two-dimensional Toeplitz because of gap-dependent corrections. A triangular Følner reduction and an exceptional-sector Gram transfer theorem extend the comparison-kernel limits to the optimal and SRM success probabilities of the full physical ensemble, yielding $p_1(c)<sup>2$. After adding a no-change hypothesis with fixed prior $π_0$, while retaining the uniform conditional distribution over anomalous intervals, the optimal joint Bayes limit is $π_0+(1-π_0)L$, where $L$ is the corresponding conditional localization limit; the weighted SRM for the augmented ensemble is not analyzed. Finite-size semidefinite programs and full dense physical-Gram SRM computations illustrate the asymptotic results.