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Conjectured continuity bound for quantum mutual information

Prove that for bipartite quantum states ρ_AB and σ_AB with trace distance δ = (1/2) ||ρ_AB − σ_AB||_1 sufficiently small, the mutual information satisfies |I(A:B)_ρ − I(A:B)_σ| ≤ log(min{|A|^2, |B|^2} − 1) + h_2(δ), where I(A:B) is the quantum mutual information and h_2 is the binary entropy function.

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Background

Motivated by their continuity results for conditional entropy under equal marginals, the authors suggest an analogous tight continuity inequality for quantum mutual information. They note that their methods do not cover this case and propose a more general conjectured inequality with explicit dimension dependence.

They further remark that, to the best of their knowledge, the conjectured inequality is open even in the classical case, underscoring its difficulty and broad interest.

References

This is not covered by Theorem~\ref{semicontinuity_relent_thm} and our proof techniques, but one might nevertheless conjecture even more generally that for states $\rho_{AB},\sigma_{AB}$ and with trace distance $\frac12 |\rho_{AB} - \sigma_{AB}|1 \leq \leq 1 - \frac{1}{|A|2}$ small enough, one has something like \left| I(A:B)\rho - I(A:B)_\sigma \right| {?} \log \Big( \min\left{|A|2,|B|2\right} - 1\Big) + h_2() In fact, to the best of our knowledge, this question seems to be open even in the classical case.

— Continuity of entropies via integral representations (2408.15226 - Berta et al., 27 Aug 2024) in Conclusion; Equation (eq:mutual-information-conj)