Brightness limits in quantum channel certification with coherent light

How much light is needed to certify that an optical channel preserves quantum information? This work shows that the answer depends on the distribution of probe brightness, even when the average photon budget is unchanged. Rare bright coherent pulses can turn an exponentially costly certification task into one with a polynomial query bound. Certification target and physical resources. The task distinguishes the identity channel from arbitrary normal, memoryless bosonic channels that are at least \(ε\) away in the energy-constrained diamond norm. This distance tests all inputs within a separate signal-energy budget, including inputs entangled with a reference. The certifier sends fresh coherent states with classical amplitude records and may use adaptive choices and quantum memory for the outputs. Every channel call, including a vacuum probe, is counted. Probe brightness is the mean photon number \(B=\|α\|^2\) of a coherent pulse. Mean energy does not determine query cost. For a fixed finite number \(m\) of modes, fixed positive probe and test-energy budgets, and fixed confidence, independent Gaussian modulation has \[log TG=Θ(ε⁻²).\] Under the same average incident-energy budget, an explicit positive rare-bright source gives \[T=O\!(ε⁻⁴ᵐ⁻⁸log1δ),\] where \(δ\) is the failure probability. The mechanism is a high-photon-number phase defect: ordinary low-brightness probes have exponentially small overlap with it, whereas a suitably weighted bright component makes it detectable while respecting the average budget. Brightness-tail resource laws. A fixed cap \(B_t≤ A\) yields the matching logarithmic order \[log Tcap=Θ\!(ε⁻²log1ε).\] For a stretched-exponential brightness budget \(∑ₜ₌₁T E eλ B_t^β≤ TM\), imposed together with the average photon constraint, the optimal orders are \[log T_β={cases}Θ(ε-2β),&0<β≤1,\\Θ\!(ε⁻²log1ε),&β>1.{cases}\] The bounds apply to the stated coherent-probe access model and are matched by physical positive sources. They quantify how restrictions on rare bright pulses change the resources needed for worst-input quantum certification. The article also develops the conversion from average coherent-probe error to a full thermal Choi-state trace-distance guarantee and to energy-constrained worst-input error, with explicit mode and energy dependence. The integrated PDF contains the concise main presentation, figures, detailed results, and complete proofs.