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Forgetting boosts the classical environment-assisted and private capacity.

David Elkouss, Sergii Strelchuk

Published Apr 27, 2016• 0 citations
OpenAlex

Abstract

A channel capacity is non-convex if the capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities. This implies that there is a concrete communication scenario in which a sender can increase the transmission rate by forgetting which channel acts on the channel input. Previously, this surprising property had only been shown for the quantum capacity. Here we prove that both the private and classical environment-assisted capacities are non-convex.

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A channel capacity is non-convex if the capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities. This implies that there is a concrete communication scenario in which a sender can increase the transmission rate by forgetting which channel acts on the channel input. Previously, this surprising property had only been shown for the quantum capacity. Here we prove that both the private and classical environment-assisted capacities are non-convex.
We study the symmetric-side-channel-assisted private capacity of a quantum channel, for which we provide a single- letter formula. This capacity is additive, convex, and, for degradable channels, equal to the unassisted private capacity. While a channel's (unassisted) capacity for private classical communication may be strictly larger than its quantum capacity, we will show that these capacities are equal for degradable channels, thus demonstrating the equivalence of privacy and quantum coherence in this context. We use these ideas to find new bounds on the key rate of quantum key distribution protocols with one-way classical post-processing. For the Bennett-Brassard-84 (BB84) protocol, our results demonstrate that collective attacks are strictly stronger than individual attacks.
For any quantum transmission line, with smaller output dimension than its input, the number of classical symbols that can be reliably encoded is strictly suboptimal. In other words, if the channel outputs a lesser number of symbols than it intakes, then rest of the symbols eventually leak into the environment, during the transmission. Can these lost symbols be recovered with minimal help from the environment? While the standard notion of environment-assisted classical capacity fails to fully capture this scenario, we introduce a generalized framework to address this question. Using an elegant example, we first demonstrate that the encoding capability of a quantum channel can be optimally restored with a minimal assistance of environment, albeit possessing suboptimal capacity in the conventional sense. Remarkably, we further prove that even the strongest two-input-two-output non-signaling correlations between sender and receiver cannot substitute for this assistance. Finally, we characterize a class of quantum channels, in arbitrary dimensions, exhibiting a sharp separation between the conventional environment-assisted capacity and the true potential for unlocking their encoding strength.
INFORMATION SCIENCE AND COMPLEX SYSTEMS SCHOOL OF SCIENCE AND TECHNOLOGY Doctor of Philosophy On environment-assisted capacities of a quantum channel by Siddharth Karumanchi We initiate the study of passive environment-assisted communication via a quantum channel, modeled as a unitary interaction between the information carrying system and an environment. In this model, the environment is controlled by a benevolent helper who can set its initial state such as to assist sender and receiver of the communication link. Here, after setting out precise definitions, focussing on the problem of quantum and classical communication, we show that entanglement plays a crucial role in this problem: indeed, the assisted capacity where the helper is restricted to product states between channel uses is different from the one with unrestricted helper. Also, prior shared entanglement between the helper and the receiver makes a difference, too. Furthermore, a scenario where there is classical communication between sender and helper (making them conferencing encoders) is discussed. We find interesting phenomenon like super-activation, super-additivity and establish non-zero lower bounds for the classical capacity with conferencing encoders.
Following initial work by Gregoratti and Werner [J. Mod. Optics 50, 913-933, 2003 and quant-ph/0403092] and Hayden and King [quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to be the harder problem). There are several, presumably inequivalent, models of the permitted local operations and classical communications between receiver and environment: one-way, arbitrary, separable and PPT POVMs. In all these models, the task of decoding a message amounts to discriminating a set of possibly entangled states between the two receivers, by a class of operations under some sort of locality constraint. After introducing the operational capacities outlined above, we show that a lower bound on the environment-assisted classical capacity is always half the logarithm of the input space dimension. Then we develop a few techniques to prove the existence of channels which meet this lower bound up to terms of much smaller order, even when PPT decoding measurements are allowed (assuming a certain superadditivity conjecture).