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: \[ρt,e,q= L1/2⊗2\![{|0000|+t^2\{(1-q)|p_+ p_+|+q|p_- p_-|\}}{1+t^2}],\] \[|p_±={|22± e(|40+|04)/√2}{√1+e^2}, 0≤ e≤14, 0≤ q≤12, E=2t^2/1+t^2.\] Here \(e\) selects the source, \(q\) is an unflagged phase-record error probability, and \(E\) is delivered mean photon number. Conversion targets \[τ_*=(18|00|+6|11|+|22|)/25, 12\|σₒᵤₜ-τ_*\|_1≤δ, 0<δ<10⁻⁵.\] Joint processing of independent supplied copies, Gaussian ancillas, squeezing, memory, Gaussian measurements, feedback, discarding and postselection are allowed. Cost is \(C= E K/p_s\): 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 \(q∈[0,1/2]\) as \(E→0\), \[log Cbest(E,q)=Θ_δ\!(max\{E-1/3,√q/E\}).\] Errors of order \(E1/3\) 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 \(b_κ\). 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^2-κ/(2-κ)/b_κ, κ/2-κ