The geometric measure of entanglement (GME) quantifies how close a multipartite quantum state is to the set of separable states under the Hilbert-Schmidt inner product. The GME can be nonmultiplicative, meaning that the closest product state to two states is entangled across subsystems. In this work, we explore the GME in two families of states: those that are invariant under bilateral orthogonal $(O\otimes O)$ transformations, and mixtures of singlet states. In both cases, a region of GME nonmultiplicativity is identified around the antisymmetric projector state. We employ state-of-the-art numerical optimization methods and models to quantitatively analyze nonmultiplicativity in these states for $d=3$. We also investigate a constrained form of GME that measures closeness to the set of real product states and show that this measure can be nonmultiplicative even for real separable states.

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• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum entanglement
• Quantum information processing
• Quantum information theory