A Geometric Analysis of Stabilizer States*

The holographic entropy cone initiates a geometric description of multipartite entanglement by defining a state's entropy vector, constructed from the ordered tuple of subsystem entanglement entropies. Necessary satisfaction of appropriate entropy inequalities constrains the position of valid entropy vectors, those representing holographically realizable states, to the interior of a convex, polyhedral holographic entropy cone, a strict subspace of the ambient 2ⁿ-1 dimensional entropy vector space. The set of stabilizer states, those quantum states reachable from vacuum through combinations of the Hadamard, Phase, and CNOT operations, constitutes a strict superset of the holographic states and similarly admit a classical dual and complete characterization of entropy within the stabilizer entropy cone. While extensive analysis of these entropy cones has returned insight into the nested structure of entropy vector subspaces, much is unknown about the transition from one class of states to another. Notably, the set of stabilizer states that are not holographically realizable indicates the existence of some non-geometric process that ejects a system from the space of holographic states, into the space of stabilizer states. Continued examination of stabilizer states reveals discerning connections between distinct classes of quantum states, illustrating the foundations of entanglement in certain quantum systems. In this work, we present a new way to analyze and classify stabilizer states and stabilizer evolution within a graph-theoretic framework and exhibit the utility of this geometric analysis in studying entanglement structures arising in high-energy physics.