The photon cost of detecting weak optical dephasing

How many uses of an optical channel are required to detect weak phase noise when only the actual mean number of submitted photons is bounded? This paper determines the optimal fixed-confidence order, including every causal protocol with quantum memory and feedback. The two hypotheses are ideal transmission and independent Gaussian phase diffusion of known variance \(\gamma>0\). In the photon-number basis the noisy channel is \[\mathcal G_\gamma(|n\rangle\langle\ell|)=e^{-\gamma(n-\ell)^2/2}|n\rangle\langle\ell|.\] A protocol has at most \(T\) channel calls. Under each hypothesis separately, the sum of unconditional mean photons submitted over all calls is at most \(TE\). Reused photons are charged on every submission, including failed or discarded branches. Inputs may be arbitrary normal states with infinite photon tails, entangled across calls and with retained reference systems. Let \(T_\delta^{\mathrm{causal}}(E,\gamma)\) be the minimum call count for both decision errors to be at most a fixed \(0<\delta<1/2\). For \(0<\gamma\le1\) and \(0<E\sqrt\gamma\le1/4\), \[T_\delta^{\mathrm{causal}}(E,\gamma)=\Theta_\delta\!\left((E\sqrt\gamma)^{-1}\right).\] The explicit bounds are \[\frac{\sqrt2[1-2\sqrt{\delta(1-\delta)}]}{E\sqrt\gamma}\le T_\delta^{\mathrm{causal}}(E,\gamma)\le\left\lceil\frac{\log(1/\delta)}{b_0E\sqrt\gamma}\right\rceil,\qquad b_0=\tfrac34(1-e^{-1/2}).\] A common-environment overlap bound charges each loss of fidelity to the two actual hypothesis-dependent photon budgets, giving the converse even with causal feedback. Independent vacuum–Fock superpositions with \(n=\lceil\gamma^{-1/2}\rceil\), photon probability \(p=E/n\), and immediate binary measurements attain the order. The powers match at fixed confidence; the numerical constants and vanishing-error dependence are distinct. The paper also separates information measures under the same mean-only constraint. The maximum single-use Umegaki relative-entropy disturbance, allowing any reference on a separable Hilbert space, is of order \(s\log(1/s)\) for \(s=E\sqrt\gamma\downarrow0\) in the stated regime. The corresponding energy-constrained Belavkin–Staszewski divergence is infinite. The entropy statement concerns single-use disturbance, while the detection theorem covers complete causal strategies. Vacuum–Fock probes and their actual photon accounting are inherited from established dephasing-discrimination work; the matching causal converse and reusable overlap and fractional-moment controls are developed with their full resource conditions. Retained memory and controller complexity are unrestricted. The integrated 11-page PDF contains two figures, complete proofs and eight references.