Classical communication over a quantum interference channel (1102.2624v5)

Abstract: Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of "very strong" and "strong" interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of "very strong" and "strong" interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates.

Classical Communication over Quantum Interference Channels

This paper explores the classical communication capacities of quantum interference channels, presenting a comprehensive exploration that generalizes classical interference channel theory to quantum systems. The quantum interference channel, characterized by two classical inputs and two quantum outputs, forms the foundation of this paper. The authors focus on channels where interference is structured as either "very strong" or "strong," drawing parallels with classical scenarios known in information theory. The crux of the paper lies in applying quantum information theory methods, particularly leveraging innovative decoding strategies, to unveil achievable communication rate regions and capacity bounds.

Introduction to Quantum Interference Channels

The paper begins by highlighting the complexity of interference channels in classical information theory—a persistent challenge due to their dual-sender, dual-receiver configuration where transmissions can interference with each other. Existing solutions, such as the Han-Kobayashi coding strategy, though effective, do not solve the general problem to its full depth. This paper progresses into the quantum domain, introducing a quantum interference channel, which retains the dual communication setup, but involves quantum outputs. The exploration primarily restricts itself to classical inputs to simplify presentation and analysis.

Key Contributions

1. Capacity in Very Strong Interference: The paper extends classical findings to quantum systems, providing a characterization of the channel's capacity when interference is "very strong." This scenario is reminiscent of Carleial's classical results, where interference is utilized rather than negated.
2. Capacity under Strong Interference: The paper employs quantum simultaneous decoding strategies to articulate the exact capacity for channels exhibiting "strong" interference. Here, a two-sender quantum simultaneous decoder is pivotal, allowing both receivers to decode both messages reliably.
3. Han-Kobayashi Quantum Rate Region: A significant hypothetical generalization is presented through a quantized Han-Kobayashi strategy, proposing a rate region achievable up to a conjectured existence of a three-sender quantum simultaneous decoder. Such a strategy underpins the capacity in scenarios where interference does not fit neatly into very strong or strong categories.
4. Outer Capacity Bounds: By iterating over diverse formulations of capacity regions, the paper also presents outer bounds akin to classical solutions, enabling a comparative perspective with existing interference channel models.
5. Connection to Unitary Gates: Finally, the paper links the quantum interference channel's findings to the capacity of bipartite unitary gates, presenting them as analogs in bidirectional communication scenarios. This association extends the relevance of the paper beyond straightforward interference channels.

Implications and Future Directions

The research provided in this paper opens pathways for understanding quantum communication complexities and extends foundational classical theory into quantum realms. Practically, these findings can influence communication protocols in quantum networks, where interference is not only inevitable but can be capitalized upon using quantum principles. The theoretical implications suggest a broader scope for quantum network analysis, potentially serving as paradigms for complex quantum information systems.

Future research, as indicated by the authors, should aim to confirm the conjecture regarding the three-sender simultaneous decoder. A validated theory in this respect would fortify the Han-Kobayashi achievable region construction. Furthermore, exploring quantum interference channels with quantum inputs might shed light on whether quantum preprocessing or entanglement offers substantial improvements over classical strategies.

In conclusion, this paper contributes to the intricate understanding of quantum communication, setting a stage for potential breakthroughs in both theory and application within multi-user quantum information systems. Its blend of theoretical innovation and alignment with classical paradigms marks a significant step forward in quantum information science.