An Entropic Lens on Stabilizer States

The holographic entropy cone provides a geometric description of multipartite entanglement by ascribing to each quantum state an entropy vector, the ordered set of von Neumann entropies of the 2ⁿ-1 reduced density matrices formed by tracing out each tensor product built from the factorized Hilbert space. Necessary satisfaction of associated entropy inequalities constrains the entropy vectors representing holographically-realizable states to the interior of a convex, polyhedral cone, a strict subspace of the ambient 2ⁿ-1 dimensional entropy vector space. In this way, entropy vectors offer a classification of states in a tensor product Hilbert space, however this alone does not select any one class as special. One way to distinguish unique states is by fixing a preferred multiplicative group of Hermitian operators to act on the Hilbert space. Stabilizer states can be defined as those quantum states reachable from vacuum through only combinations of Hadamard, Phase, and CNOT operations. This class of quantum states constitutes a strict superset of the holographic states and similarly admits a classical dual and entropic characterization through the stabilizer entropy cone. Analysis of the entropy cone model has given insight into the structure of nested entropy vector subspaces, however much is unknown about transitions from one class of states to another. We initiate a research program to combine these two classifications of states, within a graph-theoretic framework, by constructing a set of stabilizer graphs colored by entropy vector. We present new insight gained by considering restricted graphs generated by a subset of Clifford gates, specifically Hadamard and CNOT gates. At higher qubit number, we find entropy vectors which do not represent holographic states and describe their role in these stabilizer graphs. We demonstrate how higher-qubit structures can be understood as lifts of lower-qubit structures, and note the termination of new subgraphs at four qubits.