Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding (1109.0802v1)

Abstract: In this paper, we study the power of sequential decoding strategies for several channels with classical input and quantum output. In our sequential decoding strategies, the receiver loops through all candidate messages trying to project the received state onto a `typical' subspace for the candidate message under consideration, stopping if the projection succeeds for a message, which is then declared as the guess of the receiver for the sent message. We show that even such a conceptually simple strategy can be used to achieve rates up to the mutual information for a single sender single receiver channel called cq-channel henceforth, as well as the standard inner bound for a two sender single receiver multiple access channel, called ccq-MAC in this paper. Our decoding scheme for the ccq-MAC uses a new kind of conditionally typical projector which is constructed using a geometric result about how two subspaces interact structurally. As the main application of our methods, we construct an encoding and decoding scheme achieving the Chong-Motani-Garg inner bound for a two sender two receiver interference channel with classical input and quantum output, called ccqq-IC henceforth. This matches the best known inner bound for the interference channel in the classical setting. Achieving the Chong-Motani-Garg inner bound, which is known to be equivalent to the Han-Kobayashi inner bound, answers an open question raised recently by Fawzi et al. (arxiv:1102.2624). Our encoding scheme is the same as that of Chong-Motani-Garg, and our decoding scheme is sequential.

• The paper demonstrates a sequential decoding strategy for classical-quantum channels that achieves the Han-Kobayashi inner bound for the two-sender, two-receiver quantum interference channel (ccqq-IC).
• It adapts classical sequential decoding techniques using typical projectors and gentle measurement lemmas for cq-channels and ccq-MACs, leveraging geometrical properties of subspaces.
• This work extends classical interference channel results to the quantum realm, providing theoretical implications for multiuser quantum communication and potential practical impact on quantum system design.

Sequential Decoding and the Han-Kobayashi Inner Bound for Quantum Channels

The paper presented explores the capabilities of sequential decoding strategies for channels accepting classical input and producing quantum output. The author, Pranab Sen, highlights the application of these strategies to attain the Han-Kobayashi inner bound for the quantum interference channel, a key result that addresses open questions in quantum information theory.

The paper first explores the mechanics of sequential decoding techniques traditionally applied to classical channels, illustrating how these can be employed for quantum channels with classical input and quantum output (cq-channels). The receiver systematically checks candidate messages, projecting the received state onto typical subspaces correlated with these messages. If successful, the candidate is declared as the transmitted message. This decoding process is applied to achieve communication rates approaching mutual information for single sender and receiver channels and extends to multiple access channels (MAC) with two senders and one receiver (ccq-MAC).

A notable contribution of Sen's work is the decoding scheme devised for the ccq-MAC, leveraging conditionally typical projectors based on geometrical relationships between subspaces. This builds a foundation to overcome obstacles in quantum information sharing involving multiple users. Successfully overcoming these challenges enables the formulation of a sequential decoding approach that matches the known classical Chong-Motani-Garg inner bound for a two-sender, two-receiver interference channel (ccqq-IC). The equivalence of this bound to the Han-Kobayashi inner bound solidifies the success of transferring classical interference results into the quantum field.

In laying out this framework, the paper navigates the complex terrain of classical-quantum systems, incorporating facts from quantum mechanics and information theory — such as typical projectors, asymptotic smoothing, and gentle measurement lemmas — to manage the intricate interplay between typical subspaces for message candidates. The work explores the geometric factorizability of subspaces and applies it to decoupled senders' inputs in multi-access settings, thereby resolving non-trivial interference issues.

Theoretical implications lie in expanding quantum Shannon theory through more efficient multiuser communication scenarios, which may influence further advancements in quantum network theory. Practically, the results could impact the design of quantum communication systems where achieving the full potential of interference management is critical.

Future research directions may explore refining these strategies to accommodate systems with more senders or integrating advanced quantum states and entanglement into communication frameworks. Understanding these multi-dimensional interactions in quantum domains stands promising for both enhancing the theory and developing applied solutions in quantum communication technologies.