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arXiv:2609.31609

Exact series formulas for the capacities of the amplitude damping channel

Stefano Pirandola

quant-phmath-phphysics.opticsSubmitted Sep 25, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.

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We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.
Abstract: We derive an improved upper bound on the two-way-assisted capacities of the qubit amplitude-damping channel with decay probability γ\gammaγ, given by log⁡2[1+(1−γ)γγ/(1−γ)]\log_2[1+(1-\gamma)\gamma^{\gamma/(1-\gamma)}]log2​[1+(1−γ)γγ/(1−γ)]. It applies to quantum communication, entanglement distribution, private communication, and secret-key agreement, and strictly improves the PLOB relative-entropy bound, the analytic max-relative-entropy bound, and the optimized balanced squashed-entanglement bound. At high damping, a known triple-rail protocol attains more than 99.54%99.54\%99.54% of its leading coefficient. Our method extends sector teleportation simulation to qubits: phase rotations and permutations give exact simulations within fixed-excitation subspaces while preserving the two-level input structure. We determine the exact, additive relative entropy of entanglement of the amplitude-damping resources and the exact regularized entanglement gain at fixed input population. We then derive explicit bounds for generalized amplitude damping, which includes thermal excitation. Combining the amplitude-damping gain with unital and entanglement-breaking channels gives a closed expression that recovers the zero-temperature result and vanishes throughout the entanglement-breaking region. It improves established benchmarks over broad parameter ranges, while the exact entanglement of the generalized resources remains open. Both results admit refinements under an average-excitation constraint. The bounds provide quantitative limits for adaptive communication through relaxing qubits and show how a sector simulation can remain useful beyond channels whose resource entanglement is known exactly.
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The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of great interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.
In order to efficiently put quantum technologies into action, we must know the characteristics of the underlying quantum systems and effects. An interesting example is the use of the secret-key-agreement capacity of a quantum channel as a guide and measure for the implementation of quantum key distribution (QKD) and distributed quantum computation. We define the communication task of establishing a secret key over a quantum channel subject to an energy constraint on the input state and while allowing for unlimited local operations and classical communication (LOCC) between a sender and receiver. We then use the energy-constrained squashed entanglement to bound the capacity of the channel for secret-key agreement, and we show that a thermal state input maximizes a relaxation of this bound for phase-insensitive, single-mode Gaussian channels. We also establish improved upper bounds on the energy-constrained secret-key-agreement capacity for a bosonic thermal channel that is not entanglement breaking. We then generalize our results to the multipartite setting and show that the energy-constrained multipartite squashed entanglement bounds the LOCC-assisted private capacity region for a quantum broadcast channel. Next, we define the broadcast amplitude damping channel. In the setting of QKD, we discuss a communication task using the broadcast amplitude damping channel and give bounds on its achievable rate region.