Candidate pure-loss second-order converse
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- Topics
Problem
Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint? Let \(\mathcal N_\eta\) be the single-mode pure-loss channel with transmissivity \(0<\eta<1\), defined in the Heisenberg picture by
where the environment mode \(E\) is in the vacuum. In an \(n\)-use code, the channel in Eq. (1) is used to transmit one of \(M\) input states \(\rho_m^{A^n}\), with a decoding POVM \(\{\Lambda_m^{B^n}\}_{m=1}^M\). Write \(\overline\rho_{A^n}:=M^{-1}\sum_m\rho_m^{A^n}\) and let \(\Pi_{\lceil nN_S\rceil}\) project onto the \(n\)-mode subspace of total photon number at most \(\lceil nN_S\rceil\). For fixed \(N_S>0\), \(\varepsilon\in(0,1)\), and \(c>0\), impose
Let \(M^*_{\rm occ}(n,\eta,N_S,\varepsilon,c)\) be the largest \(M\) satisfying Eq. (2). Define the thermal entropy and its entropy variance by
where \(0\log_2 0:=0\). With the functions in Eq. (3), is the following upper bound valid as \(n\to\infty\)?
Here \(\Phi^{-1}\) in Eq. (4) is the inverse standard-normal cumulative distribution function, and the implicit constant may depend on \(\eta,N_S,\varepsilon,\) and \(c\), but not on \(n\).
Source
Wilde, Renes, and Guha leave open whether their Gaussian second-order expression is an upper bound and observe that such a converse should impose a photon-number occupation constraint similar to Wilde and Winter’s. Equation (2) adopts a particular exponentially strong version of that suggestion, so Eq. (4) is a precise strengthened formulation rather than a verbatim conjecture from their paper [WRG16].
Progress
Yue Tu and Yifan Jing showed that Eq. (4) fails for every \(0<\eta<1\), \(N_S>0\), \(c>0\), and \(0<\varepsilon<\tfrac12\). Each codeword carries exactly \(L=\lceil nN_S\rceil\) photons in a condensate \(\lvert u;L\rangle=(\sum_{j=1}^n u_j\hat a_j^\dagger)^L\lvert0\rangle/\sqrt{L!}\) along a unit direction \(u\in\mathbb C^n\), so the occupation constraint in Eq. (2) holds with \(\delta_n=0\). Pure loss maps \(\lvert u;L\rangle\) to a binomial mixture of the pure condensates \(\lvert u;k\rangle\), and \(\langle u;k\vert w;k\rangle=\langle u\vert w\rangle^k\). Random directions, photon counting, and a square-root measurement in each photon-number sector give
\begin{equation} \log_2 M^*_{\rm occ}(n,\eta,N_S,\varepsilon,c) \geq ng(\eta N_S)+\sqrt{n V_{\rm occ}}\,\Phi^{-1}(\varepsilon)-O(\log n), \qquad V_{\rm occ}=\frac{1-\eta}{1+\eta N_S}\,v(\eta N_S). \tag{5} \end{equation}Since \(V_{\rm occ}<v(\eta N_S)\) and \(\Phi^{-1}(\varepsilon)<0\) for \(\varepsilon<\tfrac12\), Eq. (5) exceeds the right side of Eq. (4) by a positive multiple of \(\sqrt n\). A photon-counting projector bound gives the matching converse, so under this constraint the dispersion is \(V_{\rm occ}\) rather than \(v(\eta N_S)\). The disproof is formally verified in Lean 4 with Mathlib; its only modeling input is the standard Fock-basis Kraus form of the pure-loss channel. The matching converse and the \(O(\log n)\) remainder are proved on paper only, in the proof note [TJ26].
Wilde, Renes, and Guha proved the corresponding second-order achievability expression
\begin{equation} \log_2 M^*(\mathcal N_\eta^{\otimes n},N_S,\varepsilon) \geq ng(\eta N_S) +\sqrt{n\,v(\eta N_S)}\,\Phi^{-1}(\varepsilon) +O(\log n). \tag{6} \end{equation}Their basic derivation of Eq. (6) uses a mean-photon-number ensemble. They also modify the construction to obtain exponentially small leakage outside the cutoff subspace, but with degraded second-order parameters; neither argument supplies Eq. (4) [WRG16].
Wilde and Winter proved a first-order strong converse at the rate \(g(\eta N_S)\) under the maximum-photon-number occupation constraint. For every fixed energy and rate backoff, their modified coherent-state codes have exponentially small cutoff leakage. This does not establish the exact \(g(\eta N_S)\) first-order term and the stated dispersion while also maintaining an arbitrary prescribed exponent \(c\) in Eq. (2). Moreover, a first-order strong converse controls rates a fixed distance above capacity and does not determine the \(\sqrt n\) coefficient in Eq. (4) [WW14].
Under only a mean-photon-number constraint, Wilde and Winter constructed codes with nonvanishing success probability at rates above \(g(\eta N_S)\). Hence a converse of the form Eq. (4) is false under that weaker constraint; the occupation hypothesis in Eq. (2) is essential [WW14].
Comment
Equation (4) is settled negatively for every \(0<\varepsilon<\tfrac12\) by the fixed-photon codes above. For \(\varepsilon>\tfrac12\) the stated bound is valid but not sharp, and at \(\varepsilon=\tfrac12\) the second-order terms vanish; both statements rely on the paper converse. The mean-photon-number ensemble of [WRG16] is therefore not second-order optimal under the occupation constraint. No external peer review or historical-priority claim is asserted [TJ26].
References
- [WRG16]
- M. M. Wilde, J. M. Renes, and S. Guha, “Second-Order Coding Rates for Pure-Loss Bosonic Channels,” Quantum Information Processing 15, 1289–1308 (2016).DOIarXiv
- [WW14]
- M. M. Wilde and A. Winter, “Strong Converse for the Classical Capacity of the Pure-Loss Bosonic Channel,” Problems of Information Transmission 50, 117–132 (2014).DOIarXiv
- [TJ26]
- Y. Tu and Y. Jing, “Fixed-photon condensate codes and the pure-loss second-order converse,” note for QIQCOP Zoo (23 September 2026), with a Lean 4 formalization. Proof note; Lean verification.linklink
Contributors
- Yue TuUniversity of Michigan
- Yifan JingUniversity of Michigan