Quantum Change Interval: Exact Asymptotics for Minimum Error Localization

Abstract

We study a returning quantum change interval in which a source emits $\lvert\psi\rangle$ 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 F{\o}lner 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 $c=\lvert\langle0\vert\psi\rangle\rvert$ and $p_1(x)=4(1-x^2)K^2(x^2)/\pi^2$, 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 $\tau_n=n(1-c_n)(\log n)^2$. Uniformly for $0\leq\tau_n\leq T$, we obtain $M_nP_X=(1+2\sqrt{\tau_n}/\pi+\sqrt{2}\tau_n/\pi^2)^2+O_T(\log\log n/\log n)$, 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.