Continuity of Quantum Entropic Quantities via Almost Convexity

• Published in: IEEE transactions on information theory Vol. 69; no. 9; pp. 5869 - 5901
• Main Authors: Bluhm, Andreas, Capel, Angela, Gondolf, Paul, Perez-Hernandez, Antonio
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.09.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Belavkin-Staszewski relative entropy; Concavity; Continuity; continuity bounds; Convexity; Entropy; Hilbert space; Indexes; Information theory; Mathematical Physics; Mutual information; Physics; Quantum phenomena; Quantum Physics; Quantum state; Temperature measurement; Testing; Umegaki relative entropy
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2023.3277892

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.