发表机构
School of Sciences, Hebei University of Science and Technology; Network Management Center, China Mobile Communications Group Hebei Co., Ltd.（河北科技大学理学院; 中国移动通信集团河北有限公司网络管理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对平稳独立输出序列中的瞬态纯态变化，利用集体测量推导了精确渐近定位的最优成功概率，结合平方根测量（SRM）等方法得到相关极限结果并通过数值计算验证。
AI 中文摘要
我们研究在平稳的独立输出序列中，占据一个非空连续区间的瞬态、校准纯态变化的精确标签最小误差定位，允许任意集体测量。设c=|⟨0|ψ⟩|固定，当序列长度增长时，对于每个固定的已知区间长度i，当允许平移的数量N趋于无穷大时，对应的托普利茨符号给出了渐近最优成功概率的精确平方根积分，且平方根测量（SRM）达到相同极限。若已知长度iₙ与Nₙ=n-iₙ+1均发散，且它们的比值无限制，则最优与SRM成功概率收敛于有效复合重叠c²处的一维托普利茨泛函，即p₁(c²)。在所有非空区间的均匀先验下，未知长度的物理格拉姆核因依赖间隙的修正而不是全局二维托普利茨。三角Følner约化与例外区格拉姆转移定理将比较核极限扩展到整个物理系综的最优与SRM成功概率，得到p₁(c)²。在加入具有固定先验π₀的无变化假设，同时保留异常区间的均匀条件分布后，最优联合贝叶斯极限为π₀+(1-π₀)L，其中L是对应的条件定位极限；未分析增广系综的加权SRM。有限大小的半定规划与全稠密物理-格拉姆SRM计算验证了渐近结果。
英文摘要
We study a returning quantum change interval in which a source emits $\lvertψ\rangle$ over one interval and $\lvert0\rangle$ 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 $c=\lvert\langle0\vertψ\rangle\rvert$ 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 $P_{\mathrm{opt}}(G_{N,i})-P_{\mathrm{SRM}}(G_{N,i})=O_{i,c}(N^{-1/2})$. 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 $0\leqτ_n\leq T$, we obtain $M_nP_X=(1+2\sqrt{τ_n}/π+\sqrt{2}τ_n/π^2)^2+O_T(\log\log n/\log n)$, where $X\in\{\mathrm{tr},\mathrm{SRM},\mathrm{opt}\}$. More generally, if $c_n$ approaches one from below and $n p_1(c_n)\to\infty$, the same three quantities satisfy $P_X\sim p_1(c_n)^2$. These asymptotic laws also extend to joint detection and exact localization in the presence of a no change prior.
Comments14 pages, 3 figures
