Low-energy Holevo additivity for a squeezed thermal attenuator

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

\begin{equation} E_{\mathrm{thr}}=\frac{e^{2|r|}-1}{2} +\frac{(1-\eta)(b+1/2)}{\eta}\sinh(2|r|). \tag{1} \end{equation}

For an integer \(m\geq1\), set \(H_m=\sum_{j=1}^m a_j^\dagger a_j\). Define the unrestricted Holevo information by

\begin{equation} \begin{aligned} \chi_m(E)&=\sup_{\{p_x,\rho_x\}:\,\operatorname{Tr}\bar\rho H_m\leq mE} \left[S(\Phi_{\eta,b,r}^{\otimes m}(\bar\rho)) -\sum_xp_xS(\Phi_{\eta,b,r}^{\otimes m}(\rho_x))\right],\\ \bar\rho&=\sum_xp_x\rho_x,\qquad S(\rho)=-\operatorname{Tr}\rho\log_2\rho. \end{aligned} \tag{2} \end{equation}

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

Reports do not certify correctness or automatically change the problem's status. Progress policy.

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.

References

[SKG+13]: J. Schäfer, E. Karpov, R. García-Patrón, O. V. Pilyavets, and N. J. Cerf, “Equivalence Relations for the Classical Capacity of Single-Mode Gaussian Quantum Channels,” Physical Review Letters 111, 030503 (2013).DOI arXiv
[GHG15]: V. Giovannetti, A. S. Holevo, and R. García-Patrón, “A Solution of Gaussian Optimizer Conjecture for Quantum Channels,” Communications in Mathematical Physics 334, 1553–1571 (2015).DOI arXiv
[Ji26]: S.-W. Ji, “Gaussian Optimality of Energy-Constrained One-Shot Communication through Single-Mode Bosonic Gaussian Channels,” arXiv preprint, version 2 (8 September 2026).arXiv

Low-energy Holevo additivity for a squeezed thermal attenuator

Problem

Source

Progress

Comment

References

Cite this problem

BibTeX

Plain text

Problem

Source

Progress

Comment

References

Related problems