Quantum Change Interval: Exact Asymptotics for Minimum Error Localization
Abstract
We study a returning quantum change interval in which a source emits lvertψ⟩ over one interval and lvert0⟩ elsewhere. A collective measurement on the full sequence identifies both endpoints with minimum error. We analyze the Gram matrix using Toeplitz comparison and Følner transfer, together with an exact decomposition by excitation number and interval hull. The resulting bounds establish asymptotic Bayes optimality of the square root measurement (SRM). Let c=lvert⟨0|ψ⟩rvert and p1(x)=4(1−x2)K2(x2)/π2, where K is the complete elliptic integral of the first kind. For a known interval length i, the SRM success probability and the Bayes optimum converge to the same Toeplitz symbol integral as the number N of translations grows. For fixed i and 0<c<1, their gap is Popt(GN,i)−PSRM(GN,i)=Oi,c(N−1/2). If i and N both diverge, their common limit is p1(c2), with no constraint on their relative growth. For unknown length, the uniform prior over all Mn=n(n+1)/2 nonempty intervals gives the common limit p1(c)2 at fixed overlap. For a varying overlap cn, set τn=n(1−cn)(log n)2. Uniformly for 0≤τn≤ T, we obtain MnPX=(1+2√(τn)/π+√2τn/π2)2+OT(loglog n/log n), where X∈{tr,SRM,opt}. More generally, if cn approaches one from below and n p1(cn)→∞, the same three quantities satisfy PX∼ p1(cn)2. These asymptotic laws also extend to joint detection and exact localization in the presence of a no change prior.
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BibTeX
@misc{chen2026quantum,
title = {{Quantum Change Interval: Exact Asymptotics for Minimum Error Localization}},
author = {Xu Chen and Xue Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24543},
eprint = {2608.24543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24543},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Change Interval: Exact Asymptotics for Minimum Error Localization
AU - Xu Chen
AU - Xue Ma
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.24543
UR - https://arxiv.org/abs/2608.24543
AB - We study a returning quantum change interval in which a source emits lvertψ⟩ over one interval and lvert0⟩ elsewhere. A collective measurement on the full sequence identifies both endpoints with minimum error. We analyze the Gram matrix using Toeplitz comparison and Følner transfer, together with an exact decomposition by excitation number and interval hull. The resulting bounds establish asymptotic Bayes optimality of the square root measurement (SRM). Let c=lvert⟨0|ψ⟩rvert and p_1(x)=4(1−x^2)K^2(x^2)/π^2, where K is the complete elliptic integral of the first kind. For a known interval length i, the SRM success probability and the Bayes optimum converge to the same Toeplitz symbol integral as the number N of translations grows. For fixed i and 0<c<1, their gap is P_opt(G_N,i)−P_SRM(G_N,i)=O_i,c(N^−1/2). If i and N both diverge, their common limit is p_1(c^2), with no constraint on their relative growth. For unknown length, the uniform prior over all M_n=n(n+1)/2 nonempty intervals gives the common limit p_1(c)^2 at fixed overlap. For a varying overlap c_n, set τ_n=n(1−c_n)(log n)^2. Uniformly for 0≤τ_n≤ T, we obtain M_nP_X=(1+2√(τ_n)/π+√2τ_n/π^2)^2+O_T(loglog n/log n), where X∈{tr,SRM,opt}. More generally, if c_n approaches one from below and n p_1(c_n)→∞, the same three quantities satisfy P_X∼ p_1(c_n)^2. These asymptotic laws also extend to joint detection and exact localization in the presence of a no change prior.
ER -