Classical Simulation of Quantum Channel Families: Unbounded Cost Separation and Sharp Noise–Error Scaling

When the input measurement is fixed across a channel family, the simulation of binary output tests allows test-dependent responses, whereas preparing a quantum output requires one normalized state per recorded branch. We prove an unbounded separation between the optimized outcome costs of these tasks for families admitting a common mother measurement. The channel choices are specified by classical descriptions. For an explicit family of qubit channels, six outcomes suffice for all binary tests, whereas the exact channel cost is finite at every positive level of additional depolarizing noise and diverges as the noise vanishes. The family is the image of an explicit CPTP processor with a fixed finite-dimensional program and qubit input and output. For noise fraction λ and uniform output-channel diamond-norm error ε, the minimum common-measurement cost has the sharp order (λ + ε)^{−2/3} for 0 < λ + ε ≤ 1/2, both when channels are approximated independently and when the approximating family is generated by a single CPTP processor on the same spaces. The lower bound holds for arbitrary approximating channels subject only to the common-measurement budget. At each sufficiently small positive noise level, one finite common mother serves an entire full-dimensional process neighborhood of radius proportional to λ, throughout which the same cost law and six-outcome binary-test simulator persist.