How many classical bits buy how much quantum capacity? Feedback-metered limits of lossy quantum communication

Sending quantum information over long distances is ultimately limited by photon loss. The rate limit of a lossy bosonic channel of transmissivity η is the two-way assisted capacity -log2(1-η) (the PLOB bound), whose achievability rests on unlimited classical feedback: without feedback the quantum capacity vanishes for all η ≤ 1/2, so classical back-communication acts as an on/off switch for quantum communication. Classical bits are priced for entanglement transformations and, for a fixed bipartite state, the one-way distillation trade-off between classical rate and ebit yield has a regularized entropic characterization; but no framework prices the feedback that assists a channel's quantum capacity — feedback that may steer subsequent channel inputs — and no such trade-off has been evaluated for any bosonic channel. Here we show that quantum communication can be metered in classical bits. Writing C(η,B) for the capacity of the pure-loss channel under a feedback budget of B classical bits per channel use, we construct the first achievable-rate frontier C_{ach}(η,B) ≤ C(η,B) and prove the onset law C_{ach}(η,B) ≥ ηlog2(m_B)/m_B, m_B = ⌈ H_2(η)/B⌉, valid for every B>0 with envelope (η/H_2(η)) Blog2(H_2(η)/B) as B → 0 — an exact identity R = η Blog2(1/B) under fixed-length accounting, so the Blog(1/B) order is accounting-convention invariant. The B=0 switch thereby resolves, at the level of achievable rates, into a continuous but non-analytic onset: even for η ≤ 1/2 any positive budget buys a positive quantum rate, each bit unlocking ~log2(1/B) ebits at small budgets and refuting the naive pricing C ≤ Q + B. At large budgets a metered quantum-state-merging family, consuming backward one-way classical communication only, achieves C_{ach}(η,B) ≥ PLOB - [eη^2/(2ln 2)] 2^{-B}(1+o(1)): each backward bit per use halves the achievable distance to PLOB, which exposes the 73.6% plateau of the purification families as a family artifact. Standard entanglement-measure converses are provably blind to B, so what remains open is only strict saturation at finite B, now confined to an exponentially small window. The frontier is also a design tool: on a network of lossy links, the joint allocation of per-link feedback budgets and photon directions under per-site classical budgets is a convex program, and on a 20-site fibre network its optimum doubles the end-to-end entanglement rate relative to a fixed orientation. An appendix extends the frontier to multimode loss, where Grassmann-quantized beamforming controls the PLOB loss at rate 2^{-B/(M-1)}. Preprint (companion paper: Conservation laws for the classical communication cost of one-way entanglement distillation, with entropic converse bounds for the pure-loss channel). Ancillary files: reproduction scripts and data; code maintained at https://github.com/deeptell-inc/fmplob.