Sharp quantum conditional-entropy continuity bound

The problem asks for the optimal uniform bound on changes in $H(A|B)$ controlled by trace distance and $d=\dim A$. The unrestricted fully quantum case had remained open after classical, quantum-classical, and equal-$B$-marginal cases were settled.

Known results

• Alicki and Fannes established an earlier dimension-dependent continuity estimate; Winter strengthened it.
• Alhejji and Smith determined the sharp classical modulus in 2020.
• Wilde proved the quantum-classical result and conjectured the fully quantum expression.
• Berta, Lami, and Tomamichel, and independently Audenaert, Datta, Jabbour, Capel, and Gondolf, proved the equal-$B$-marginal case.

July 2026 sharp-bound theorem

Theorem 1.1 gives, for $T(\rho_{AB},\sigma_{AB})\leq\delta$ and $d=\dim A\geq2$, the optimal bound $h_2(\delta)+\delta\log(d^2-1)$ up to $\delta=1-d^{-2}$, followed by $2\log d$. It is attained for every $\delta\in[0,1]$ when $\dim B\geq d$, via maximally entangled/Weyl-sector examples. The proof idea was developed with assistance from ChatGPT 5.6 Sol.

Current status (as of July 2026): the unrestricted sharp bound and its optimality are settled by the arXiv theorem; no part of the stated problem remains open.