Conservation laws for the classical communication cost of one-way entanglement distillation, with entropic converse bounds for the pure-loss channel

Entanglement distillation over the pure-loss bosonic channel is governed by the PLOB bound, -log2(1-η) ebits per use, but this benchmark presumes unlimited two-way classical communication. When the classical feedback budget is metered at B bits per use, the achievable rate approaches PLOB exponentially in B, yet no matching converse is known, and we argue that the standard route — static entanglement monotones — cannot work even in principle. Here we develop a converse program built on the Devetak–Harrow–Winter (DHW) characterization of one-way distillation. Our central result is a conservation law: for every rank-one instrument applied by the sending party, the single-copy communication charge R and entanglement yield E obey the exact identity R+E=S(B), the local entropy of the measured system, independent of the state; general instruments obey R+E=S(B)-S(E'|B'X), with a correction that vanishes for all projective coarse-grainings. Combining the identity with the proven quantum entropy power inequality yields an unconditional lower bound on the Bob-entropy cost of approaching PLOB, and, for the class of sharp instruments satisfying a per-instrument yield condition, a two-sided bound η^2 2^{-B} lesssim PLOB-C lesssim (e/2) η^2 2^{-B}: in this class the feedback-metered capacity provably falls short of PLOB at every finite B, with the gap determined up to a factor e/2. For two-mode-squeezed inputs — and, non-asymptotically, for the whole class of products of two-mode-squeezed states of arbitrary energies dressed by passive linear optics — we prove a complementary converse that needs no yield condition at all: a separable Gaussian witness pins the relative entropy of entanglement of the loss output to within Theta(1/bar N) of PLOB, reproducing the 2^{-B} gap scaling unconditionally for these input classes, with constants matched to within 0.78–0.94 of the achievability side for two-mode-squeezed inputs (smaller for the passive class at η ≥ 0.5). We further identify the obstruction to removing the yield condition in general: measurements can increase the reverse coherent information by up to their heralding information, E ≤ I_R + I(X;E), and we exhibit state families saturating this and show numerically that the excess is generic even for loss-channel outputs. The sharp-class converse is thereby reduced to a single variational problem, which we state precisely and conjecture to stay below PLOB; beyond the sharp class, we show the conservation law acquires a coarse-graining discount that must be bounded separately. Preprint (converse-side companion of: How many classical bits buy how much quantum capacity? Feedback-metered limits of lossy quantum communication). Ancillary files: reproduction scripts and data; code maintained at https://github.com/deeptell-inc/fmplob.