Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique (2207.08791v4)

Abstract: We consider a quasi-classical version of the Alicki-Fannes-Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called "quasi-classical". Several applications of the proposed method are described. Among others, we obtain the universal continuity bound for the von Neumann entropy under the energy-type constraint which in the case of one-mode quantum oscillator is close to the specialized optimal continuity bound presented recently by Becker, Datta and Jabbour. We obtain semi-continuity bounds for the quantum conditional entropy of quantum-classical states and for the entanglement of formation in bipartite quantum systems with the rank/energy constraint imposed only on one state. Semi-continuity bounds for entropic characteristics of classical random variables and classical states of a multi-mode quantum oscillator are also obtained.

• The paper presents near-optimal continuity bounds for von Neumann entropy using a quasi-classical Alicki-Fannes-Winter technique.
• It refines methods for quantum conditional entropy and entanglement of formation, improving analysis under energy-type constraints.
• The work extends continuity bounds to multi-mode oscillators and discrete classical systems, offering practical insights for quantum technologies.

Analysis of Continuity Bounds in Quantum Information Theory

The paper by M.E. Shirokov provides a comprehensive exploration of continuity bounds for von Neumann entropy and related entropic characteristics within quantum information theory. Utilizing a quasi-classical version of the Alicki-Fannes-Winter technique, the paper establishes continuity bounds across quantum states that exhibit certain quasi-classical attributes. This work extends the existing methodologies by focusing on systems where traditional approaches may not suffice, specifically within the context of what the author defines as "quasi-classical" states.

The core contribution of the paper is in refining the continuity bounds for the von Neumann entropy and other entropic measures under various constraints. The refinement provided by these bounds is close to optimality, particularly when compared with recent advancements in the field—such as those by Becker, Datta, and Jabbour for one-mode quantum oscillators. Shirokov's treatment includes:

1. Universal Continuity Bounds: The paper proposes continuity and semi-continuity bounds for the von Neumann entropy that advance those proposed by Winter. These include bounds close to optimal for quantum oscillators where only one mode is considered, extending to incorporate energy-type constraints.
2. Quantum Conditional Entropy: The research further explores providing continuity bounds for quantum conditional entropy, especially for quantum-classical states; it provides sharper bounds when traditional energy constraints are applied only to part of the system.
3. Entanglement of Formation: Shirokov utilizes the refinement of continuity bounds to advance the understanding of the entanglement of formation criteria under constraints, demonstrating improvements over existing measures by using standard techniques and new quasi-classical interpretations.
4. Characteristics of Random Variables: The work develops continuity bounds for information-theoretic quantities of random variables. This includes the mutual information and conditional entropy within discrete classical systems, aligning well with the established quantum paradigms.
5. Multi-mode Quantum Oscillators: By investigating classical states of multi-mode quantum oscillators, the paper successfully extends the methodology for calculating entropic characteristics, ensuring that the bounds accommodate quantum systems traditionally difficult to analyze under existing paradigms.

The implications of this research are significant in the broader context of quantum information theory. It showcases that by incorporating the framework of quasi-classical systems, it is possible to derive stronger and more general continuity bounds that can potentially be applied in practical quantum computing, quantum communication, and other quantum technologies. Moreover, these findings emphasize the nuances introduced by quasi-classical settings, offering fresh perspectives and methodologies for tackling long-standing problems in quantum thermodynamics and resource theory.

Looking forward, these insights pave the way for substantial developments in the mathematical underpinnings of quantum theories and could inspire new approaches in understanding complex quantum systems. The continuity bounds serve as a critical tool in the ongoing discourse on entropy and information, possibly impacting the formulation of new quantum algorithms and communication protocols. This work is a detailed and methodological expansion of the toolkit available for researchers and practitioners in quantum information science, etching out paths for future explorations into entropic measures in both classically intuitive and quantum-specific domains.