Wilde’s conjecture: tight continuity bound for the quantum conditional entropy

Determine whether, for all finite-dimensional bipartite quantum states ρ_AB and σ_AB with trace distance δ = (1/2) ||ρ_AB − σ_AB||_1 ≤ 1 − 1/|A|^2, the conditional entropy satisfies the bound |H(A|B)_ρ − H(A|B)_σ| ≤ log(|A|^2 − 1) + h_2(δ), where H(A|B) denotes the quantum conditional entropy and h_2 is the binary entropy function.

Background

The paper develops a general semi-continuity inequality for relative entropy and uses it to prove a tight continuity bound for the quantum conditional entropy in the special case where the two states have the same marginal on the conditioning system. This result matches the form conjectured by Wilde but only under the equal-marginals assumption.

Wilde’s original conjecture posited a fully quantum uniform continuity bound for the conditional entropy analogous to the classical and quantum-classical results. While this paper settles the conjectured form when the conditioning marginals coincide, the general case without this restriction remains unresolved and is identified as the main open question.