Low-energy Holevo additivity for a squeezed thermal attenuator
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- Topics
Problem
Is energy-constrained Holevo information additive for a squeezed thermal attenuator below its water-filling threshold? Fix \(0<\eta<1\), \(b\geq0\), and \(r\in\mathbb R\setminus\{0\}\). The channel \(\Phi_{\eta,b,r}\) mixes an input mode with an independent centered squeezed thermal mode on a beam splitter of transmissivity \(\eta\). Each use has a fresh environment. With \([q,p]=i\), \(a=(q+ip)/\sqrt2\), and \(R=(q,p)^T\), use \(V_{jk}=\langle\{R_j-\langle R_j\rangle,R_k-\langle R_k\rangle\}\rangle/2\). The environment covariance is \((b+1/2)\operatorname{diag}(e^{2r},e^{-2r})\). Choose an energy \(0<E<E_{\mathrm{thr}}\), where
For an integer \(m\geq1\), set \(H_m=\sum_{j=1}^m a_j^\dagger a_j\). Define the unrestricted Holevo information by
The states in Eq. (2) may be entangled across uses. Continuous ensembles are allowed, with sums replaced by integrals. Does \(\chi_m(E)=m\chi_1(E)\) hold for every \(m\) throughout the regime in Eq. (1)?
Source
This is the squeezed-attenuator specialization of the low-energy additivity gap discussed by Schäfer et al., around Corollary 2 and their conclusion [SKG+13]. Ji’s version 2 explicitly retains the multiple-use gap in its Scope paragraph and Supplemental Material, Sec. 13 [Ji26].
Progress
For \(E\geq E_{\mathrm{thr}}\), a Gaussian distribution of displaced squeezed-vacuum letters attains the thermal maximum-output-entropy bound minus the minimum output entropy. The resulting value per use is \(g_2(\eta E+(1-\eta)[(b+1/2)\cosh(2r)-1/2])-g_2((1-\eta)b)\), where \(g_2(x)=(x+1)\log_2(x+1)-x\log_2x\) and \(0\log_2 0=0\). Corollary 2 of Schäfer et al. gives the threshold construction; the subsequent Gaussian optimizer theorem supplies the unrestricted minimum-entropy result [SKG+13], [GHG15].
Ji’s September 2026 preprint, version 2, Theorem 2, claims \(\chi_1(E)=\chi_1^{\mathrm G}(E)\) at every energy. Here the Gaussian restriction uses displaced pure Gaussian letters with Gaussian modulation. The theorem is explicitly one-use and does not establish the equality for \(m>1\) [Ji26].
The entanglement-breaking regime \(b\geq\eta/(1-\eta)\) is additive at every energy. Squeezing equivalence preserves entanglement breaking, and the energy-constrained additivity theorem applies to the resulting Gaussian channel. See Sec. 3, Eqs. (20)–(24), of Giovannetti, Holevo, and García-Patrón [GHG15].
Comment
The question allows arbitrary block ensembles and fixes the average photon budget per use. The unsettled regime is below threshold and outside the entanglement-breaking region. One-use Gaussian optimality does not settle regularization. The cited 2026 claim is a preprint result. Squeezing the channel into phase-insensitive form changes the input cost, so phase-insensitive capacity additivity alone does not resolve this question.