Quantum conditional mutual information (QCMI) measures the correlation between two quantum systems A and B given knowledge of a third system C. It's always non-negative — the strong subadditivity of von Neumann entropy — and it's zero if and only if the state is a quantum Markov chain. These properties make it the fundamental measure of quantum correlations conditioned on side information.
The paper provides exact characterizations of QCMI and related entropies — not inequalities but equalities that express these quantities in terms of operationally meaningful objects. The conditional mutual information I(A:B|C) equals the minimum relative entropy between the state and the set of quantum Markov chains with the same C-marginal. This is not an inequality (which was known) but an exact identity: the QCMI is precisely the distance to the nearest Markov chain.
Similar exact characterizations are derived for the conditional entropy, the coherent information, and other entropic quantities. Each characterization expresses the entropy as a variational quantity — an optimization over a specific set of states or channels — where the optimum is achieved and the optimal point has a physical interpretation.
The characterizations convert abstract entropic quantities into concrete optimization problems. The conditional mutual information is no longer just “the amount of correlation between A and B that survives conditioning on C” — it's “the minimum cost of making the state Markov.” The number has a construction behind it, not just a formula.