Inner bounding the quantum entropy cone with subadditivity and subsystem coarse grainings

We show via explicit construction that all the extreme rays of both the three-party quantum entropy cone and the four-party stabilizer entropy cone can be obtained from subsystem coarse grainings of specific higher-party quantum states, namely, extreme states characterized by saturating a (nontrivial) maximal set of instances of subadditivity. This suggests that the study of the “subadditivity cone,” and the set of its extreme rays realizable in quantum mechanics, provides a powerful tool for deriving inner bounds for the quantum and stabilizer entropy cones, as well as constraints on new inequalities for the von Neumann entropy.

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A hypergraph giving an element of ${\mathcal{R}}_6$. The hyperedge (yellow blobs) has weight 2, and edges (blue lines) have weight 1.

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A hypergraph giving an element of ${\mathcal{R}}_9$. The hyperedge (yellow blobs) has weight 2, and edges (blue lines) have weight 1.

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A hypergraph giving an element of ${\mathcal{R}}_8$. The two hyperedges (yellow and purple blobs) have weight 2, and edges (blue lines) have weights either 1 (when unspecified) or 2.