A Loss-Limited Optimum for Fixed Uniform Multi-Pass Quantum Sensing per Absorbed Photon

Multi-pass schemes send a photon through a sample several times to learn more about it. When the sample rather than the light is scarce, the natural figure of merit is the information gained per photon the object absorbs. We show that three properties fix the loss-limited optimum of every fixed scheme in which a single photon passes repeatedly and uniformly through the sample and is detected once at the end: a payoff that grows as the square of the pass number, the same survival probability on every pass, and no dose from a photon already lost. They force a single trade-off function h(x)=x2/(ex−1), where x is the number of passes times the loss per pass. Its maximum, 0.648, sits at xopt=1.594. The multi-pass optima of Koppell and Kasevich and of Yu et al. are two instances, and the loss-limited ceiling of interaction-free interrogation is a third when its payoff is counted as conclusive detections. Phase sensing of a weakly absorbing object is optimal at mopt=xopt/(ϵ+α) passes; the absorption α and the parasitic loss ϵ enter the optimum only through their sum, but the damage counts only α. N00N states and their loss-robust generalisations do worse per absorbed photon, and optimising over photon number and pass number together returns a single recycled photon; Gaussian probes and chains tailored pass by pass lie outside this class and can exceed its ceiling by a bounded factor. Two further problems are priced in the same measure. Absorption estimation gains nothing, for any scheme. Above the crosstalk crossover of the mode sorter, detecting a faint companion below the Rayleigh limit costs the companion a dose that does not depend on its faintness; below it, and in direct imaging at any separation, the dose grows without bound as the companion fades. Among single photons and path-entangled number states, the gentlest measurement of a fragile sample is also the simplest one: a single photon, recycled.