The conditional quantum mutual information (CQMI) of a state $\rho^{ABC}$ respects the dimension bound $I(A:B|C)_{\rho}{\leq}2 \log_2 \{\min (d_A, d_B)\}$ (Mark Wilde's book, exercise 11.7.9). One example of a state saturating the bound is when the systems $A$ and $B$ are maximally entangled. In that case, the CQMI reduces to quantum mutual information, $I(A:B|C)_{\rho}{=}I(A:B)_{\rho}=2 \log_2 \{\min (d_A, d_B)\}$.

Do we have a more general example? More precisely, what is the necessary and sufficient condition to saturate the bound?



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