Optimal purity estimation under correlated phase drift and photon loss
Year: 2026
Type: preprint
Abstract: Purity does not change under a phase rotation, but estimating it from successive optical pulses can still require relative-phase information. This article determines the best possible precision when an unknown single-mode source is independently prepared, undergoes photon loss, and is then rotated by a hidden correlated phase trajectory. A uniform precision law. The normal source state satisfies \(\operatorname{Tr}(\rho N)\le E\) and \(\operatorname{Tr}(\rho N^s)\le M\), with fixed \(E,M>0\) and \(s\ge1\). The phase starts uniformly on the circle, independently of the entire sequence of independent increments, whose known Fourier spectrum is \(\exp(-\gamma|q|^\beta)\), with \(\gamma>0\) and \(0<\beta\le2\). The target is the purity \(P_\eta=\operatorname{Tr}[\mathcal L_\eta(\rho)^2]\) after pure loss, not the purity of the phase-averaged marginal. For \(m\) source preparations, set \(p=1-\eta\), \(L_m=[\log(m+\mathrm e)]^{1/\beta}\) and \(q_m=\lceil[\log m/(2\gamma)]^{1/\beta}\rceil\). The minimax expected absolute error obeys \[ R_m(P_\eta)\asymp F_m(p),\qquad F_m(p)=L_m^{-s}\eta^{q_m}+L_m^{-2s}(1+pL_m)^{-2s}. \] The comparison constants are uniform for \(\eta\in[1/2,1]\), including transmissivities approaching one as the budget grows. An ideal adjacent-pair interference measurement, independent of the transmissivity, attains the order. The converse allows arbitrary joint measurements on the complete correlated quantum history. The same law holds for the second Rényi entropy \(-\log P_\eta\). What loss changes. Vacuum–high-photon coherence and coherence between broad high-photon packets give two different hard pairs of legal sources. Without loss the rate is \((\log m)^{-s/\beta}\); at fixed nonzero loss it is \((\log m)^{-4s/\beta}\). Each loss level has its own output target. The faster rate therefore does not mean improved recovery of the original input purity. Both constructions retain their actual moment budgets, the same source within each experiment and complete photon tails. Mean cost, stopping and unbiased readings. Allowing arbitrary quantum memory, feedback and an observable stopping time leaves the order unchanged when only the worst-case mean number of requested raw source slots is bounded by \(B\): \[ R_B^{\mathrm{stop}}(P_\eta)\asymp F_{\lfloor B\rfloor}(1-\eta). \] Every requested slot is charged, including discarded slots and failed outcomes. An exactly unbiased purity estimator attains the same absolute-error order with finite mean cost and almost-sure termination, using rare, unbounded signed readings. Under the specified independent common-phase reset and auxiliary-reset permissions for repeated blocks, exact unbiasedness and uniformly finite mean cost rule out uniformly finite variance. This is a uniform obstruction over sources, not a claim of infinite variance at every source; exact unbiasedness is not asserted for the Rényi entropy. The analysis builds on swap/parity measurements, drift diagnostics, quadratic-functional deconvolution and randomized telescoping. Its contribution is the matching quantum-history law, its uniform loss dependence and its return to the actual mean preparation budget. The integrated 14-page PDF contains two figure groups, complete proofs and references. The figures illustrate asymptotic rate functions, not experimental data or calibrated sample requirements; detector, phase-control, storage and elapsed-time costs are separate from the source-slot optimum.
Author: He, Zixuan
Institution: University of Glasgow
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225629677
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