Candidate pure-loss second-order converse
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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
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{5} \end{equation}Their basic derivation of Eq. (5) 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
The unresolved task is the candidate sharp Gaussian \(\sqrt n\) converse term under the stated fixed-exponent occupation constraint, not the first-order capacity or strong converse. No matching achievability theorem is known for this exact constraint. The exclusion of \(\eta=1\) is essential: at the identity-channel endpoint, the dimension of the cutoff Fock subspace itself violates Eq. (4) for \(\varepsilon<\tfrac12\).