Quantum Change Interval: Exact Asymptotics for Minimum Error Localization

Abstract

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.