Continuity of quantum entropic quantities via almost convexity

Abstract

Based on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.

68 pages, 6 figures. This is a long version of the paper arXiv:2305.10140. v3: We corrected a previous flaw in Lemma 5.8

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Cited by in corpus (14)

• Quantum Rényi and $f$-divergences from integral representations
• Continuity bounds on observational entropy and measured relative entropies
• General Continuity Bounds for Quantum Relative Entropies
• Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
• Quantum processes as thermodynamic resources: the role of non-Markovianity
• Observational entropy with general quantum priors
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Unified framework for continuity of sandwiched Rényi divergences
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Transcendental properties of entropy-constrained sets: Part II
• Asymptotically Optimal Tests for One- and Two-Sample Problems
• New monotonicity property of the quantum relative entropy
• Lower semicontinuity of the relative entropy disturbance and its corollaries