Coding Theorem and Strong Converse for Quantum Channels (1409.2536v1)

Abstract: In this correspondence we present a new proof of Holevo's coding theorem for transmitting classical information through quantum channels, and its strong converse. The technique is largely inspired by Wolfowitz's combinatorial approach using types of sequences. As a by-product of our approach which is independent of previous ones, both in the coding theorem and the converse, we can give a new proof of Holevo's information bound.

• The paper introduces a new proof of Holevo's theorem using maximal code arguments instead of random coding methods.
• The paper demonstrates that exceeding a quantum channel's capacity sharply reduces transmission success, establishing a strong converse.
• It adapts combinatorial techniques from classical information theory to extend the analysis to infinite alphabets and constrained encodings.

Revisiting Holevo's Coding Theorem and Strong Converse for Quantum Channels

The paper by Andreas Winter presents a significant advancement in the understanding of classical information transmission through quantum channels, primarily revisiting and providing a new proof for Holevo's coding theorem alongside its strong converse. This work employs techniques inspired by Wolfowitz's combinatorial approach, which leverages the method of types, a well-established concept in classical information theory, reinterpreted in the quantum context. This novel approach diverges from previous attempts which relied heavily on adapting random coding methods for quantum states.

Key Contributions

1. New Proof of Holevo's Theorem: Winter's paper offers a new proof of Holevo's theorem using maximal code arguments instead of the random coding methods previously adapted for quantum states. This approach provides an independent, elementary proof of the Holevo bound, which is central to understanding the channel capacity for transmitting classical information over quantum channels.
2. Strong Converse: The paper effectively proves the strong converse of Holevo's theorem, asserting that the probability of successful information transmission drops sharply if the transmission rate exceeds the quantum channel's capacity. This insight adds depth to the theoretical frameworks governing information limits in quantum communication channels.
3. Extension to Infinite Alphabets: The techniques devised are adaptable to situations involving encoding under linear constraints and scenarios with infinite input alphabets. This adaptability broadens the applicability of the paper's results beyond typical finite-dimensional counterparts.
4. Elementary Proof Technique: The paper demonstrates that an elementary proof strategy, closely related to classical methods, can effectively establish fundamental results in quantum information theory, thus bridging a gap between classical and quantum domains.

Technical Highlights

• Typical Projectors and Shadows: A critical component of Winter's methodology is the use of typical projectors and their associated shadows. These concepts allow one to capture the essence of sequence behavior in quantum states, akin to type and typical set concepts in classical information theory.
• Measurement Disturbance: The paper quantitatively formulates the intuition that measurements with high probability of success minimally disturb the measured quantum state, a crucial insight for developing efficient coding strategies in quantum information.
• Capacity Estimation: By leveraging typical projections and variance-typical sequences, a new perspective on data compression and channel capacity for quantum channels is provided. This method helps to reinforce the capacity estimation methodologies under quantum settings.

Implications and Future Directions

The implications of this research are profound for both theoretical and practical applications in quantum communications. By providing a robust theoretical foundation that aligns closely with well-understood classical methods, Winter enables the exploration of new quantum communication strategies that could lead to more efficient coding schemes and error-correction techniques.

Practically, understanding and leveraging quantum channels' capacity limits is vital for the development of quantum networks and the burgeoning field of quantum computing. The work suggests future research paths in exploring coding theorems under even broader conditions, potentially incorporating different quantum states and channel characteristics.

Moreover, by suggesting that simpler measurement techniques might suffice in certain quantum channel coding situations, this work indirectly invites further exploration into physical implementations of such strategies that could optimize quantum communication systems.

In conclusion, this paper reinforces the vitality of foundational theoretical work in quantum information theory, providing robust tools and perspectives that can guide both theoretical refinements and practical innovations in quantum communication technologies.