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arXiv:1903.07747 · Physical Review A

Information-theoretic aspects of the generalized amplitude-damping channel

Sumeet Khatri, Kunal Sharma, Mark M. Wilde

Published Jul 6, 2020• 74 citations
arXivHTMLPDF

Abstract

The generalized amplitude-damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analog of the bosonic thermal channel, and it thus can be used to model lossy processes in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is antidegradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in Rosati et al., [Nat. Commun. 9, 4339 (2018)] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.

DOI
10.1103/physreva.102.012401

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arXiv:1912.00931•IEEE Transactions on Information Theory•Mar 24, 2021•22 citations
The generalized amplitude-damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analog of the bosonic thermal channel, and it thus can be used to model lossy processes in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is antidegradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in Rosati et al., [Nat. Commun. 9, 4339 (2018)] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.
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.
The classical capacity of a quantum channel generally requires regularization because input codewords may be entangled across channel uses. We remove this regularization for generalized amplitude-damping channels (GADCs) and determine their exact strong-converse exponent throughout the full parameter range. The exponent is a single-letter transform of the sandwiched Rényi Holevo information and is attained by product input codewords with collective decoding. The key ingredient is a decomposition of the joint output into positive operators whose triangular factors have identical scalar Gram matrices after norm compression. A sharp triangular Schatten-norm compression inequality then establishes maximum output norm multiplicativity for diagonally sandwiched GADCs tensored with arbitrary completely positive maps. This yields additivity of the sandwiched Rényi Holevo information at every order above one. Consequently, the classical capacity equals the known one-copy Holevo information and admits a scalar optimization. Thus, entangled input codewords improve neither the capacity nor the optimal exponential decay of decoding success above capacity.
The generalized amplitude damping channel (GADC) is considered an important model for quantum communications, especially over optical networks. We make two salient contributions in this paper apropos of this channel. First, we consider a symmetric GAD channel characterized by the parameter n=1/2,n=1/2,n=1/2, and derive its exact classical capacity, by constructing a specific induced classical channel. We show that the Holevo quantity for the GAD channel equals the Shannon capacity of the induced binary symmetric channel, establishing at once the capacity result and that the GAD channel capacity can be achieved without the use of entanglement at the encoder or joint measurements at the decoder. Second, motivated by the inevitable buffering of qubits in quantum networks, we consider a generalized amplitude damping \emph{queue-channel} -- that is, a setting where qubits suffer a waiting time dependent GAD noise as they wait in a buffer to be transmitted. This GAD queue channel is characterized by non-i.i.d. noise due to correlated waiting times of consecutive qubits. We exploit a conditional independence property in conjunction with additivity of the channel model, to obtain a capacity expression for the GAD queue channel in terms of the stationary waiting time in the queue. Our results provide useful insights towards designing practical quantum communication networks, and highlight the need to explicitly model the impact of buffering.
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.
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.