arXiv:1110.5746 · Physical Review A

Private and quantum capacities of more capable and less noisy quantum channels

Shun Watanabe

Published Jan 24, 2012• 50 citations

Abstract

Two classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.

DOI: 10.1103/physreva.85.012326

Related papers

Remarks on Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

Shun Watanabe

Oct 26, 20112 citations

Bounding Quantum Capacities via Partial Orders and Complementarity

Christoph Hirche, Felix Leditzky

Quantum capacities are fundamental quantities that are notoriously hard to compute and can exhibit surprising properties such as superadditivity. Thus, a vast amount of literature is devoted to finding tight and computable bounds on these capacities. We add a new viewpoint by giving operationally motivated bounds on several capacities, including the quantum capacity and private capacity of a quantum channel and the one-way distillable entanglement and private key of a quantum state. These bounds are generally phrased in terms of capacity quantities involving the complementary channel or state. As a tool to obtain these bounds, we discuss partial orders on quantum channels and states, such as the less noisy and the more capable order. Our bounds help to further understand the interplay between different capacities, as they give operational limitations on superadditivity and the difference between capacities in terms of the information-theoretic properties of the complementary channel or state. They can also be used as a new approach towards numerically bounding capacities, as discussed with some examples.

arXiv:2202.11688IEEE Transactions on Information TheoryAug 17, 202212 citations

Potential Capacities of Quantum Channels

Andreas Winter, Dong Yang

We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their plain analogues. We thus study upper bounds on some potential capacities. For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first lift the channel to a Hadamard channel and then prove that the quantum and private capacity of a Hadamard channel is strongly additive, implying that for these channels, potential and plain capacity are equal. Employing these upper bounds, we show that if a channel is noisy, however close it is to the noiseless channel, then it cannot be activated into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the so-called environment-assisted quantum capacity, because we are able to characterize its potential version.

arXiv:1505.00907IEEE Transactions on Information TheoryJan 21, 201637 citations

Quantum and Private Capacities of Low-Noise Channels

Felix Leditzky, Debbie Leung, Graeme Smith

We determine both the quantum and the private capacities of low-noise quantum channels to leading orders in the channel's distance to the perfect channel. It has been an open problem for more than 20 yr to determine the capacities of some of these low-noise channels such as the depolarizing channel. We also show that both capacities are equal to the single-letter coherent information of the channel, again to leading orders. We thus find that, in the low-noise regime, superadditivity and degenerate codes have a negligible benefit for the quantum capacity, and shielding does not improve the private capacity beyond the quantum capacity, in stark contrast to the situation when noisier channels are considered.

arXiv:1705.04335Physical Review LettersApr 20, 201846 citations

Private communication via zero-private-capacity quantum channels

Chengkai Zhu, Xin Wang

Private communication over a noisy quantum channel requires reliable transmission to the receiver and secrecy from the environment. Whether two channels with zero private capacity can jointly enable private communication is a longstanding open problem in quantum information theory. Here we resolve this problem by exhibiting a four-level channel and a qubit erasure channel with half erasure probability, each with zero private capacity, whose joint use achieves more than 0.0001903 private bits per product use. The encoding gives the receiver a linear information gain with at most quadratic environmental leakage, enabling privacy through a fixed joint measurement and classical coding. This superactivation, impossible for independent classical memoryless wiretap channels, shows that a channel's private capacity alone does not determine its value for secure communication. The initial activation example was identified through interactions with large language models, and the result has been formalized in Lean 4.

quant-pharXiv:2609.10520Sep 10, 20260 citations

Optimizing the fundamental limits for quantum and private communication

Xin Wang

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.

Dec 2, 20196 citations