Quantum capacity of a bosonic thermal attenuator

Problem

Let \(\Phi_{\eta,\nu}\) be the single-mode bosonic thermal attenuator with transmissivity \(0<\eta<1\) and environmental mean photon number \(0<\nu<\infty\), defined by

\begin{equation} \Phi_{\eta,\nu}(\rho_A) :=\operatorname{Tr}_{E'}\!\left[ U_\eta(\rho_A\otimes\tau_{\nu,E})U_\eta^\dagger \right], \qquad \tau_{\nu,E}:=\sum_{k=0}^{\infty} \frac{\nu^k}{(\nu+1)^{k+1}} |k\rangle_E\!\langle k|_E, \tag{1} \end{equation}

where \(U_\eta:AE\to BE'\) is a beam-splitter unitary and \(\tau_{\nu,E}\) acts on the environment mode \(E\). For an environment mode of angular frequency \(\omega_E\) at temperature \(T\), \(\nu=(e^{\hbar\omega_E/(k_{\rm B}T)}-1)^{-1}\). Thus \(0<T<\infty\) is equivalent to \(0<\nu<\infty\) for fixed \(\omega_E>0\); \(\nu=0\) corresponds to \(T=0\), while \(\nu\to\infty\) as \(T\to\infty\). Its unassisted quantum capacity is the regularized coherent information

\begin{equation} \mathcal Q(\Phi_{\eta,\nu}) :=\lim_{n\to\infty}\frac1n\sup_{\rho_{A^n}} I_{\rm c}(\rho_{A^n},\Phi_{\eta,\nu}^{\otimes n}), \qquad I_{\rm c}(\rho,\mathcal N) :=S(\mathcal N(\rho))-S(\mathcal N^{\rm c}(\rho)), \tag{2} \end{equation}

where \(S(\sigma):=-\operatorname{Tr}(\sigma\log_2\sigma)\) and \(\mathcal N^{\rm c}\) is any complementary channel. Determine Eq. (2) for every \(0<\eta<1\) and \(0<\nu<\infty\), equivalently every strictly positive finite environment temperature.

Source

Rosati, Mari, and Giovannetti explicitly study the unknown thermal-attenuator quantum capacity through nonmatching narrow bounds; the exact-capacity problem is retained after the later non-Gaussian separation [RMG18], [MCF+26].

Progress

At zero environmental temperature, the pure-loss capacity is

\begin{equation} \mathcal Q(\Phi_{\eta,0}) =\left[\log_2\frac{\eta}{1-\eta}\right]_+, \qquad [x]_+:=\max\{0,x\}. \tag{3} \end{equation}

Thus Eq. (3) solves only the boundary case \(\nu=0\) [WPG07].
Optimizing the one-use coherent information over all single-mode Gaussian input states gives exactly

\begin{equation} \sup_{\rho\in\mathsf G_1}I_{\rm c}(\rho,\Phi_{\eta,\nu}) =\left[\log_2\frac{\eta}{1-\eta}-g(\nu)\right]_+, \qquad g(x):=(x+1)\log_2(x+1)-x\log_2x. \tag{4} \end{equation}

Holevo and Werner computed the thermal-input limit, Brádler proved that the thermal-state supremum is attained asymptotically at infinite input energy, and Mele et al. proved that squeezing cannot improve it within \(\mathsf G_1\). Equation (4) is nevertheless not optimal over arbitrary non-Gaussian inputs [HW01], [Br15], [MCF+26].
The exact antidegradability region is

\begin{equation} \eta\leq\eta_{\rm AD}(\nu):= \frac{\nu+\frac12}{\nu+1}, \qquad \mathcal Q(\Phi_{\eta,\nu})=0. \tag{5} \end{equation}

Hence Eq. (2) is settled throughout the region in Eq. (5) [LKA+19].
In the non-entanglement-breaking region, a bottleneck decomposition gives the analytic upper bound

\begin{equation} \mathcal Q(\Phi_{\eta,\nu}) \leq\left[ \log_2\frac{\eta-(1-\eta)\nu}{(1-\eta)(\nu+1)} \right]_+ \quad\text{when }\eta>(1-\eta)\nu. \tag{6} \end{equation}

The channel is entanglement breaking, and therefore has zero capacity, when the displayed condition fails. The bound in Eq. (6) was obtained independently by Rosati et al. and Noh et al. [RMG18], [NAJ19]. Degradable extensions and channel-concatenation orderings subsequently produced tighter parameter-dependent upper bounds, but no known upper bound is tight throughout the non-antidegradable region [FKG21], [KFG24].
Asymptotic multimode symplectic-lattice GKP codes achieve a rate whose gap from the upper bound in Eq. (6) is at most \(\log_2 e\simeq1.443\) qubits per channel use in the energy-unconstrained setting. This is a constant-gap coding theorem, not an exact capacity formula [NAJ19].
The latest rigorous lower bounds use non-Gaussian one-mode inputs:

\begin{equation} \begin{aligned} \sup_{\rho\in\mathsf G_1}I_{\rm c}(\rho,\Phi_{4/5,1}) &=0, &\qquad \mathcal Q(\Phi_{4/5,1}) &\geq4.7\times10^{-4},\\ \mathcal Q(\Phi_{0.7841,1}) &\geq1.4312\times10^{-7},\\ \eta_{\rm c} &:=\inf\{\eta:\mathcal Q(\Phi_{\eta,1})>0\}, &\qquad 0.75\leq\eta_{\rm c} &\leq0.7841. \end{aligned} \tag{7} \end{equation}

The first certificate uses an explicit rank-two state supported on six Fock levels; numerical optimization over related families reaches about \(8.4\times10^{-3}\) qubits per use at \((\eta,\nu)=(0.8,1)\), but that larger value is not certified. The second certificate in Eq. (7) also propagates by data processing to a two-dimensional parameter region. These results disprove single-mode Gaussian optimality but do not determine the capacity [MCF+26].

Comment

For generic \(\nu>0\) outside the zero-capacity regions, the lower and upper bounds do not coincide. Non-Gaussian one-mode optimization and possible multimode superadditivity both remain relevant to Eq. (2).

References

[WPG07]

M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels,” Physical Review Letters 98, 130501 (2007).DOI arXiv

[HW01]

A. S. Holevo and R. F. Werner, “Evaluating Capacities of Bosonic Gaussian Channels,” Physical Review A 63, 032312 (2001).DOI arXiv

[Br15]

K. Brádler, “Coherent Information of One-Mode Gaussian Channels—The General Case of Non-Zero Added Classical Noise,” Journal of Physics A: Mathematical and Theoretical 48, 125301 (2015).DOI arXiv

[RMG18]

M. Rosati, A. Mari, and V. Giovannetti, “Narrow Bounds for the Quantum Capacity of Thermal Attenuators,” Nature Communications 9, 4339 (2018).DOI arXiv

[NAJ19]

K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates with Gottesman–Kitaev–Preskill Codes,” IEEE Transactions on Information Theory 65, 2563–2582 (2019).

DOI arXiv

[LKA+19]

L. Lami, S. Khatri, G. Adesso, and M. M. Wilde, “Extendibility of Bosonic Gaussian States,” Physical Review Letters 123, 050501 (2019).DOI arXiv

[FKG21]

M. Fanizza, F. Kianvash, and V. Giovannetti, “Estimating Quantum and Private Capacities of Gaussian Channels via Degradable Extensions,” Physical Review Letters 127, 210501 (2021).DOI arXiv

[KFG24]

F. Kianvash, M. Fanizza, and V. Giovannetti, “Low-Ground/High-Ground Capacity Regions Analysis for Bosonic Gaussian Channels,” International Journal of Quantum Information 22, 2440005 (2024).DOI arXiv

[MCF+26]

F. A. Mele, G. Catalano, M. Fanizza, V. Giovannetti, and L. Lami, “Bosonic Quantum Communication Beyond the Thermal Threshold,” arXiv preprint (2026).arXiv

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