Phase records control the conversion cost of non-Gaussian light

Detecting non-Gaussian light establishes that a resource is present. Its structure and preparation record determine the cost of turning it into a specified output. This paper gives matching conversion-cost laws for weak optical sources and a general input certificate that can exclude every allowed Gaussian receiver before a circuit is designed. Sources with identical energy, covariance and total photon statistics can have different cost exponents; losing a classical phase bit can make a fixed-source conversion impossible. Source, target and accounting. The Wigner-positive two-mode sources are obtained by half loss on each mode: \[\rho_{t,e,q}=\mathcal L_{1/2}^{\otimes2}\!\left[\frac{|00\rangle\langle00|+t^2\{(1-q)|p_+\rangle\langle p_+|+q|p_-\rangle\langle p_-|\}}{1+t^2}\right],\] \[|p_\pm\rangle=\frac{|22\rangle\pm e(|40\rangle+|04\rangle)/\sqrt2}{\sqrt{1+e^2}},\quad 0\le e\le\tfrac14,\quad 0\le q\le\tfrac12,\quad E=\frac{2t^2}{1+t^2}.\] Here \(e\) selects the source, \(q\) is an unflagged phase-record error probability, and \(E\) is delivered mean photon number. Conversion targets \[\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25,\qquad \tfrac12\|\sigma_{\mathrm{out}}-\tau_*\|_1\le\delta,\quad 0<\delta<10^{-5}.\] Joint processing of independent supplied copies, Gaussian ancillas, squeezing, memory, Gaussian measurements, feedback, discarding and postselection are allowed. Cost is \(C=\mathbb E K/p_s\): every source request counts, including vacuum requests and failed attempts. The task uses ideal Gaussian operations and charges supplied copies. 1. A source-selection law controlled by record accuracy. Optimizing both the source parameter \(e\) and the allowed Gaussian protocol gives, uniformly in \(q\in[0,1/2]\) as \(E\to0\), \[\log C_{\mathrm{best}}(E,q)=\Theta_\delta\!\left(\max\{E^{-1/3},\sqrt{q/E}\}\right).\] Errors of order \(E^{1/3}\) or smaller preserve the fastest logarithmic cost. Above that scale, record quality controls the optimum. Equivalently, for \(q=q_0E^a\), fixed \(0 0\), any fixed positive unflagged error thus eventually obstructs this output. Flagged erasures behave differently: independently retaining a correct bit with probability \(a_r>0\) adds at most a factor \(1/a_r\) to a protocol that discards erasures. 4. A transferable exclusion certificate and an explicit receiver. A verified positive phase-space minorant with a physical pure Gaussian curvature reference supplies a source budget \(b_\kappa\). For a requested one-mode output whose trace-distance gap from the Gaussian convex hull is at least \(g\), every allowed Gaussian protocol obeys \[C\ge\frac{g^2-\kappa/(2-\kappa)}{b_\kappa},\qquad \frac{\kappa}{2-\kappa}