Positivity threshold for thermal-attenuator quantum capacity

Problem

For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be

\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 \(T\to\infty\). Define its unassisted quantum capacity and its critical transmissivity by

\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 \eta_{\rm c}(\nu) :=\inf\{\eta\in(0,1):\mathcal Q(\Phi_{\eta,\nu})>0\}, \tag{2} \end{equation}

where \(I_{\rm c}(\rho,\mathcal N) :=S(\mathcal N(\rho))-S(\mathcal N^{\rm c}(\rho))\). Determine the exact threshold curve in Eq. (2) for the channel in Eq. (1).

Source

The exact positivity threshold is implicit in the gap between the antidegradability boundary of Lami et al. and the non-Gaussian positive-rate witnesses of Mele et al. [LKA+19], [MCF+26].

Progress

Exact antidegradability gives the lower bound

\begin{equation} \eta_{\rm c}(\nu)\geq\eta_{\rm AD}(\nu) :=\frac{\nu+\frac12}{\nu+1}. \tag{3} \end{equation}

The capacity vanishes at and below the right-hand side of Eq. (3) [LKA+19].
Thermal input states give the upper bound

\begin{equation} \eta_{\rm c}(\nu)\leq\eta_{\rm G}(\nu) :=\frac{2^{g(\nu)}}{1+2^{g(\nu)}}, \qquad g(x):=(x+1)\log_2(x+1)-x\log_2x. \tag{4} \end{equation}

Above the right-hand side of Eq. (4), the thermal-state coherent information is positive [HW01].
For one environmental thermal photon, certified non-Gaussian inputs and antidegradability narrow the threshold to

\begin{equation} \frac34\leq\eta_{\rm c}(1)\leq0.7841 <\eta_{\rm G}(1)=\frac45. \tag{5} \end{equation}

In particular, Eq. (5) proves that the Gaussian threshold is not the true positivity threshold [MCF+26].

Comment

This is the positivity-boundary subproblem of determining the full capacity in Problem 49. Even the single value in Eq. (5) is not known exactly.

References

[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
[HW01]: A. S. Holevo and R. F. Werner, “Evaluating Capacities of Bosonic Gaussian Channels,” Physical Review A 63, 032312 (2001).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

Positivity threshold for thermal-attenuator quantum capacity

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