Two-way quantum capacity of a thermal amplifier
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Problem
What is the two-way quantum capacity of a thermal amplifier for every gain and thermal noise level? Fix a power gain \(\kappa>1\) and \(b\geq0\). The single-mode thermal amplifier \(\mathcal A_{\kappa,b}\) is the Gaussian channel realized by two-mode squeezing with output annihilation operator
The independent input and environment modes obey \([a,a^\dagger]=[e,e^\dagger]=1\). The environment is a fresh independent thermal mode at each use, with mean photon number \(b\) and density operator
Here \(|n\rangle\) is the \(n\)-photon Fock state. The unused output mode is discarded. Equations (1) and (2) specify the channel on arbitrary input states. Protocols may use arbitrary adaptive local quantum operations and unlimited two-way public classical communication. The parties initially share no entanglement or secret key. There is no input-energy bound; the capacity is the supremum over finite mean input-energy budgets. The two-way quantum capacity \(Q_2\) is the supremum of asymptotic qubits transmitted per use with vanishing error. Equivalently, it is the maximal rate of maximally entangled pairs distributed with vanishing trace-distance error.
Source
Pirandola et al. give the thermal-amplifier capacity bounds in Eqs. (26)–(28) [PLOB17]. Mele, Lami, and Giovannetti retain the unresolved exact value in Supplemental Material, Sec. V.2 [MLG25].
Progress
Let \(g_2(x)=(x+1)\log_2(x+1)-x\log_2x\), with \(g_2(0)=0\). Coherent information and the PLOB converse give
\begin{equation} \max\left\{0,\log_2\frac\kappa{\kappa-1}-g_2(b)\right\} \leq Q_2(\mathcal A_{\kappa,b}) \leq\log_2\frac{\kappa^{b+1}}{\kappa-1}-g_2(b),\qquad 0\leq b<\frac1{\kappa-1}. \tag{3} \end{equation}The bounds in Eq. (3) coincide at \(b=0\), giving \(Q_2(\mathcal A_{\kappa,0})=\log_2[\kappa/(\kappa-1)]\). See Eqs. (26)–(28) and Supplementary Note 4 [PLOB17].
Supplemental Theorem S21 proves \(Q_2(\mathcal A_{\kappa,b})>0\) exactly when \(b<1/(\kappa-1)\), including under every positive finite input-energy budget. The channel is entanglement breaking outside this region. Supplemental Theorem S22 gives a recurrence-and-hashing lower bound that can exceed coherent information, so the displayed coherent-information rate is not the exact answer [MLG25].
Comment
The noisy regime below the entanglement-breaking threshold remains unresolved. This asks for entanglement transmission with two-way classical assistance. Secret-key generation and unassisted quantum transmission are distinct tasks; their capacities need not coincide with \(Q_2\) away from \(b=0\). Reverse-direction Braunstein–Kimble teleportation over channel-generated two-mode squeezed states also identifies \(Q_2(\mathcal A_{\kappa,b})=Q_2(\mathcal L_{1/\kappa,b})\). Here \(\mathcal L_{\eta,b}\) is the thermal attenuator with transmissivity \(\eta\) and the same thermal occupancy \(b\). This editorial deduction uses the covariance formula in Sec. 6, Eqs. (52)–(54), and finite-block simulation followed by entanglement hashing [LLSBP18]. It concerns the supremum over finite energy budgets, not equality at the same fixed budget.