Quantum f-divergences and error correction (1008.2529v6)

Abstract: Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

• The paper proves the monotonicity of quantum f-divergences under substochastic maps when the function is operator convex, extending prior results on distinguishability measures and reversibility theorems.
• These monotonicity findings are applied to quantum error correction, showing that maps preserving pairwise distinguishability can be reversed via a canonically derived map.
• A significant technical contribution is the integral representation of operator convex functions, serving as a foundational tool for extending results on monotonicity and equality cases.

Quantum f-Divergences and Error Correction

The paper, authored by Fumio Hiai, Milán Mosonyi, Dénes Petz, and Cédric Bény, explores the concept of "quantum f-divergences," a generalization of classical f-divergences within quantum frameworks and their applications in quantum error correction. Quantum f-divergences are intrinsically tied to Petz's quasi-entropies and serve as a cornerstone in quantum information theory, underpinning various distinguishability measures such as quantum relative entropy, Rényi relative entropies, and the Chernoff and Hoeffding distances.

Monotonicity under Substochastic Maps

The central thrust of the paper is the demonstration of the monotonicity of quantum f-divergences under substochastic maps when the defining function is operator convex. This result extends and consolidates prior findings on monotonicity for distinguishability measures. Monotonicity essentially indicates that the act of further randomizing two quantum states cannot increase their distinguishability, a concept crucial for maintaining the integrity of measurements in quantum systems. The research analyzes conditions when the monotonicity inequality holds with equality, expanding Petz's reversibility theorem for a broader class of f-divergences, adding depth to the theoretical understanding of quantum state transformation and reversibility.

Implications for Quantum Error Correction

The paper applies these monotonicity findings to quantum error correction, showing that if a stochastic map preserves pairwise distinguishability as measured by an appropriate f-divergence, its operation can be reversed via another stochastic map derived canonically from the original. This feeds into practical quantum computing strategies where error correction is vital, ensuring quantum systems can be shielded from the detrimental effects of noise and other errors.

Integral Representation and Convexity

A significant technical contribution is the integral representation of operator convex functions over the positive half-line. This provides a foundational tool for extending known results on monotonicity inequality and equality cases. This representation is noteworthy for further exploration and might find applications beyond the context of this paper.

Special Cases and Inverse Hölder Inequality

The authors also delve into specific scenarios where convexity suffices for monotonicity, departing from operator convexity, illustrating the inverse Hölder inequality for operators—a mathematical nuance that could yield new insights into quantum mechanics and quantum information theory.

Future Directions

While the paper firmly grounds several theoretical aspects of quantum f-divergences and ties them to practical applications like error correction, it opens avenues for future research. There is potential for these theoretical tools to be extended to solve broader classes of quantum information issues, including those that may arise as quantum technologies evolve.

The paper's insights consolidate existing theory while enabling practical advancements in quantum error correction, offering a rigorous foundation for further exploration in quantum information theory. Future work might aim to refine these conditions, broadening their applicability or exploring additional settings where these principles can be effectively leveraged.