The exact catalyst return cost of producing photons in a vacuum signal

How much must an optical catalyst change when it supplies photons to an initially empty signal? This article gives an exact finite-budget benchmark: a specified unconditional probability of producing a nonvacuum signal requires a minimum disturbance of the entire returned catalyst, even when the output may be correlated. Exact return-error frontier. Let the catalyst have initial mean photon number at most \(E\), while the signal and all other auxiliary ports start in vacuum. A deterministic unitary strongly conserves total photon number. Define \(a\) as the actual, unconditional probability that the output signal is nonvacuum, and \(d_C=\tfrac12\|\tau'_C-\tau_C\|_1\) as the catalyst's return error. For \(0\lt a\le1\) and \(E\ge a\), put \[ m=\left\lfloor\frac{2E}{a}-1\right\rfloor,\qquad F(E,a)=\frac{2[(m+1)a-E]}{m(m+1)}. \] Every allowed process satisfies \(d_C\ge F(E,a)\), and a finite-mode catalyst with a passive cyclic permutation attains equality for every feasible pair \((E,a)\). If \(E\lt a\), the event probability is infeasible; if \(a=0\), the minimum error is zero. The converse covers normal states with photon-number coherence, infinite photon tails, finitely or countably many modes, correlations with an untouched reference, and correlated outputs. The same lower bound holds for the joint catalyst–reference return distance. Equality is a global optimization over the catalyst, its modes, the operation and the reference association, not a guarantee for each preselected catalyst. A convenient consequence is \[ d_C\ge\frac{a^2}{2E-a},\qquad E\ge\frac a2+\frac{a^2}{2d_C}. \] This smooth bound agrees with the exact frontier at the discrete budgets \(E/a=(m+1)/2\). For certain photon production, \(a=1\), a return error of one per cent requires at least 50.5 photons of initial mean catalyst number, and the construction attains this budget. Mechanism and prior work. A common input–output number record preserves the actual source and its conservation law. Every successful event crosses a number threshold, whose lost probability is bounded by the return trace distance. The mean photon budget then limits how this event probability can be distributed across thresholds. Filling the lowest levels first gives the exact piecewise frontier. Energy-ladder shifts, energy-constrained embezzlement and the equal-weight optical queue are established ingredients; the contribution here is the all-process converse and its attained finite-budget frontier for arbitrary event probability. The matching source uses a weighted version of the known queue. What the guarantee means. For a single signal mode with prescribed negative parity \(-\operatorname{Tr}[(-1)^{N_S}\rho'_S]=s\), \(0\lt s\le1\), the exact globally optimized return cost is \(F(E,(1+s)/2)\), with feasibility requiring \(E\ge(1+s)/2\). The attaining permutation transfers negativity already present in the catalyst; it does not create that resource from a Wigner-positive source. The budget counts mean photon number, not a per-run photon cutoff or physical energy at arbitrary unequal mode frequencies. The catalyst loses at least \(a\) photons on average in every allowed process, so a small single-use trace-distance change does not imply cost-free restoration or repeated use. Populated auxiliary inputs and postselected success branches obey different resource conditions. The integrated seven-page PDF contains two figures, the complete proofs and references; its constructions and curves are theoretical, not experimental performance claims.