Sharp continuity bounds for quantum conditional entropy

How much can conditional quantum entropy S(A|B) change when two states have trace distance at most δ? The desired bound depends only on d=dim A, allowing an arbitrary finite-dimensional conditioning system B.

Why it matters and what counts as progress

Why it matters: The sharp modulus quantifies stability of conditional entropy without dependence on the size of the conditioning system.

What counts as progress: A universal sharp bound h₂(δ)+δ log(d²−1) up to δ=1−d⁻² and 2 log d thereafter, with states attaining it when dim B≥d.

How the problem developed

27 Jul 2026 · Sharp bound [1]
Mario Berta, Pablo Costa Rico, Gereon Kossmann, Ludovico Lami, and Julius A. Zeiss Proved the optimal dimension-only continuity modulus h₂(δ)+δ log(d²−1) up to δ=1−d⁻², followed by a 2 log d plateau; the bound is tight for every δ when dim B≥d.

AI contribution

Sharp continuity of quantum conditional entropy

Mario Berta, Pablo Costa Rico, Gereon Kossmann, Ludovico Lami, Julius A. Zeiss

For finite-dimensional quantum states at trace distance at most δ and d=dim A, the optimal dimension-only conditional-entropy continuity modulus is h₂(δ)+δ log(d²−1) up to δ=1−d⁻², then 2 log d. Equality is attainable when dim B≥d.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFull