Trade-off capacities of the quantum Hadamard channels

TL;DR

This paper derives exact formulas for the entanglement-assisted classical and quantum communication capacities of Hadamard channels, including dephasing, cloning, and Unruh channels, demonstrating their tractable capacity regions and improved coding strategies.

Contribution

It provides the first tractable capacity region formulas for a broad class of quantum channels, specifically Hadamard channels, and introduces a measure to evaluate coding strategy improvements.

Findings

Exact capacity formulas for Hadamard channels.

Demonstration of superior coding strategies over time-sharing.

Application to physically relevant channels like dephasing, cloning, and Unruh channels.

Abstract

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel's capabilities for transmitting classical or quantum information. Here, we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a naive time-sharing strategy, and we introduce a measure to determine this improvement.