Energy-constrained quantum capacity of a thermal attenuator
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- Topics
Problem
For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be
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 \(T\to\infty\). For a finite mean input photon number \(0<N_{\rm S}<\infty\), define the energy-constrained unassisted quantum capacity by
where \(\hat n_j\) is the photon-number operator of the \(j\)th mode and \(I_{\rm c}(\rho,\mathcal N) :=S(\mathcal N(\rho))-S(\mathcal N^{\rm c}(\rho))\). Determine Eq. (2) for the thermal channel in Eq. (1).
Source
The problem is implicit in the nonmatching energy-constrained bounds of Sharma et al. and the strict multimode achievable-rate improvements of Noh, Pirandola, and Jiang [SWA+18], [NPJ20].
Progress
For the zero-temperature pure-loss channel, the constrained capacity is known exactly:
\begin{equation} \mathcal Q(\Phi_{\eta,0},N_{\rm S}) =\left[g(\eta N_{\rm S})-g((1-\eta)N_{\rm S})\right]_+, \qquad g(x):=(x+1)\log_2(x+1)-x\log_2x. \tag{3} \end{equation}Equation (3) does not extend directly to a thermal environment [WQ18].
For \(\nu>0\), a thermal input of mean photon number \(N_{\rm S}\) gives
\begin{equation} \mathcal Q(\Phi_{\eta,\nu},N_{\rm S}) \geq[L(\eta,\nu,N_{\rm S})]_+, \tag{4} \end{equation}where the rate in Eq. (4) is
\begin{equation} \begin{aligned} L(\eta,\nu,N_{\rm S}) &:={} g(\eta N_{\rm S}+(1-\eta)\nu)\\ &\quad-g\!\left( \frac{\Delta+(1-\eta)N_{\rm S}-(1-\eta)\nu-1}{2} \right)\\ &\quad-g\!\left( \frac{\Delta-(1-\eta)N_{\rm S}+(1-\eta)\nu-1}{2} \right),\\ \Delta &:=\sqrt{[(1+\eta)N_{\rm S}+(1-\eta)\nu+1]^2 -4\eta N_{\rm S}(N_{\rm S}+1)}. \end{aligned} \tag{5} \end{equation}The expression in Eq. (5) is a one-use achievable rate, not an exact capacity formula [HW01], [SWA+18].
Multimode Gaussian constructions improve the raw thermal-input rate through the convexified bound
\begin{equation} \mathcal Q(\Phi_{\eta,\nu},N_{\rm S}) \geq\sup_{0<x\leq1} xI_{\rm c}(\tau_{N_{\rm S}/x},\Phi_{\eta,\nu}) \geq[L(\eta,\nu,N_{\rm S})]_+. \tag{6} \end{equation}For \(\nu=N_{\rm S}=1\), this construction numerically improves the single-mode thermal rate in a nontrivial loss regime [NPJ20].
A data-processing decomposition gives, for \(1/2\leq\eta<1\),
\begin{equation} \mathcal Q(\Phi_{\eta,\nu},N_{\rm S}) \leq[U(\eta,\nu,N_{\rm S})]_+, \qquad U:=g(\eta'N_{\rm S})-g((1-\eta')N_{\rm S}), \quad \eta':=\frac{\eta}{1+(1-\eta)\nu}. \tag{7} \end{equation}The upper bound in Eq. (7) generally does not meet Eq. (6) [SWA+18].
The unconstrained optimization over all single-mode Gaussian inputs is known, but its proof fixes output entropy rather than input energy and explicitly does not establish thermal-state optimality at fixed \(N_{\rm S}\) [MCF+26].
Comment
For finite \(\nu>0\) and finite \(N_{\rm S}\), neither the optimal one-use input nor the necessity of regularization is known. This constrained problem is distinct from the unconstrained capacity in Problem 49.
References
- [WQ18]
- M. M. Wilde and H. Qi, “Energy-Constrained Private and Quantum Capacities of Quantum Channels,” IEEE Transactions on Information Theory 64, 7802–7827 (2018).DOIarXiv
- [HW01]
- A. S. Holevo and R. F. Werner, “Evaluating Capacities of Bosonic Gaussian Channels,” Physical Review A 63, 032312 (2001).DOIarXiv
- [SWA+18]
- K. Sharma, M. M. Wilde, S. Adhikari, and M. Takeoka, “Bounding the Energy-Constrained Quantum and Private Capacities of Phase-Insensitive Bosonic Gaussian Channels,” New Journal of Physics 20, 063025 (2018).DOIarXiv