Fixed-photon-number codes and a proposed pure-loss converse
A zero-leakage shell-code construction exceeds the proposed occupation-constrained second-order upper bound.
Yuxuan Zhang
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Full manuscript (PDF, 4 pages) · LaTeX source, proof and checks (ZIP)
Status: a complete counterexample candidate, internally checked with AI assistance. External expert review and publication priority remain unconfirmed. This is the round-15 result, prepared for publication on 20 September 2026. It has not been formally verified by a proof assistant.
The question
The QIQCOP statement proposes a second-order upper bound on classical communication through a pure-loss bosonic channel. Codewords must place all but exponentially small average weight below a total-photon cutoff \(\lceil nN_S\rceil\). Its proposed bound is
\[\log_2 M^*\leq ng(\eta N_S)+\sqrt{nv(\eta N_S)}\,\Phi^{-1}(\varepsilon)+O(\log n),\]where \(g(x)=(x+1)\log_2(x+1)-x\log_2x\) and \(v(x)=x(x+1)[\log_2(1+1/x)]^2\). The page presents a strengthened formulation motivated by earlier coding papers. This note addresses that formulation; it does not identify a false theorem in those papers.
The candidate result
For every fixed \(0<\eta<1\), \(N_S>0\) and \(0<\varepsilon<1\), deterministic codes with exactly \(k=\lceil nN_S\rceil\) photons achieve, for every sufficiently large \(n\),
\[\begin{aligned} \log_2 M_n={}&ng(\eta N_S) +\sqrt{n\eta(1-\eta)N_S}\log_2\!\left(1+\frac1{\eta N_S}\right)\Phi^{-1}(\varepsilon)\\ &-\frac32\log_2 n+O(1). \end{aligned}\]Every input has zero cutoff leakage, so any prescribed positive leakage exponent is satisfied. For \(\varepsilon<1/2\), the negative normal quantile makes this achieved rate exceed the proposed upper bound by a positive multiple of \(\sqrt n\).
For example, \(\eta=1/2\), \(N_S=2\) and \(\varepsilon=\Phi(-1)\) give
\[\log_2 M_n=2n-\sqrt{n/2}-\tfrac32\log_2n+O(1),\]whereas the proposed converse would require \(\log_2M_n\leq2n-\sqrt{2n}+O(\log n)\). A logarithmic remainder cannot absorb that difference.
The finite-block argument
Choose independent unit vectors \(u_m\in\mathbb C^n\) and encode message \(m\) in the supermode number state
\[|u_m;k\rangle=\frac{(\sum_i u_{m,i}a_i^\dagger)^k}{\sqrt{k!}}|0\rangle.\]After loss, the surviving photon number \(L\) is binomial, and conditional output states are \(\lvert u_m;l\rangle\). Haar averaging makes these states isotropic in the \(l\)-photon sector of dimension \(d_l=\binom{n+l-1}{l}\). A Gram–Schmidt measurement has expected average error at most \((M-1)/(2d_l)\) in that sector.
One common codebook and a direct-sum POVM across sectors therefore give a deterministic code with error at most
\[\Pr(L<t)+\frac{M-1}{2d_t},\qquad L\sim\operatorname{Bin}(k,\eta).\]Choose the discrete \(\varepsilon-1/n\) quantile for \(t\) and \(M=\lfloor d_t/n\rfloor\). Berry–Esseen and Stirling estimates yield the stated rate. The full manuscript checks integer rounding, the bounded remainder, and completion to a POVM on the entire output space.
Verification and prior work
The pure-loss Kraus output, exact shell moments and finite-code decoder identities were reproduced in clean containers. The analytic proof separately covers the asymptotic claim; finite numerical examples are not its justification. The source archive preserves the corrected review packet and the initial attachment failure.
Photon-sector isotropy, passive-interferometer ensembles and binomial number-state loss are established ingredients, explicitly credited to Fanizza et al., Quantum 5, 608 (2021), §3.2 Eq. (9) and §5 Eq. (42). Wilde–Renes–Guha, §6.1, gives a coherent-state occupation-constrained construction; Wilde–Winter proves a first-order strong converse. Priority for the precise operational second-order construction remains unconfirmed.
No optimal replacement dispersion, efficient encoder or practical decoder is claimed.
Authorship and preparation
The proof, checking and exposition were developed with substantial AI assistance under the author’s direction. Internal AI reviews do not constitute independent human peer review. Scientific Agent Skills informed manuscript preparation and evidence tracking and is cited in the PDF. Primary papers retain their own terms and are not bundled in the source archive.