Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints (1507.07775v6)

Abstract: We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, $E_R$, and its regularization $E_R^\infty$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E_F^\infty$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.

• The paper refines the Alicki-Fannes bounds for quantum conditional entropy using system dimensions and trace distances.
• It applies entropy inequalities and energy constraints to restore continuity in infinite-dimensional quantum systems with bounded energy.
• These findings enhance the stability analysis of quantum systems, providing crucial tools for quantum computing and cryptography.

Uniform Continuity Bounds in Quantum Entropy

The paper by Andreas Winter presents a rigorous exploration of continuity bounds for quantum entropies, examining both finite and infinite-dimensional systems. The research primarily focuses on two classes of continuity bounds: those related to conditional von Neumann entropy and those associated with relative entropy distances from convex sets. This analysis is pivotal in quantum information theory, particularly for understanding the behavior of quantum systems under small perturbations.

Tight Continuity Bounds for Conditional Entropy

The paper revises the Alicki-Fannes continuity bounds initially established for quantum conditional entropy, which are crucial for dealing with small changes in quantum states. Winter's formulation offers nearly optimal results based solely on the system dimensions and trace distances. By providing a tighter form of these continuity bounds, the paper advances the understanding of the asymptotic continuity of quantum entanglement measures such as the entanglement of formation and its regularization. The use of entropy inequalities, including strong subadditivity, underpins the derivation of these bounds, showcasing their robustness and applicability across different scenarios.

Energy Constraints in Infinite Dimensions

A noteworthy contribution of this paper is the extension of continuity bounds to infinite-dimensional systems where energy constraints come into play. In such settings, traditional entropy measures can become discontinuous. The paper demonstrates that with bounded energy states, continuity can be reinstated, evidenced through systems comprising multiple quantum harmonic oscillators. Interestingly, in this context, the Gibbs entropy takes on a role analogous to the Hilbert space dimension in the finite-dimensional setting.

Practical Implications and Future Directions

The results have significant implications for the theoretical underpinnings of quantum information theory. By offering a foundation for understanding how small deviations in quantum states affect entropy, the paper provides a toolset for analyzing the stability of quantum systems under operational conditions. Practically, these bounds facilitate better insights into quantum entanglement, an essential resource in quantum computing and quantum cryptography. Furthermore, they can inform the strategic development of quantum technologies by providing guidelines that account for energy constraints in analysis and implementation.

As for future research avenues, extending these continuity bounds to scenarios that transcend standard convex set assumptions presents an intriguing challenge. Further exploration of these topics could unveil deeper structural insights into quantum systems and enhance the practical applicability of these theoretical constructs.

Ultimately, Winter's work contributes a fundamental piece to the quantum information puzzle, addressing continuity concerns in quantum entropy while adapting to both system size and energy constraints. Such advancements are critical for the continued evolution of quantum computation and communication, promising a more profound comprehension of entropic behavior in quantum mechanical systems.