Exploring Non-Multiplicativity in the Geometric Measure of Entanglement

• URL: http://arxiv.org/abs/2503.23247v2
• Date: Fri, 04 Apr 2025 13:35:12 GMT
• Title: Exploring Non-Multiplicativity in the Geometric Measure of Entanglement
• Authors: Daniel Dilley, Jerry Chang, Jeffrey Larson, Eric Chitambar,
• Abstract summary: The geometric measure of entanglement (GME) quantifies how close a multi-partite quantum state is to the set of separable states under the Hilbert-Schmidt inner product.<n>We explore the GME in two families of states: those that are invariant under bilateral $(O otimes O)$ transformations, and mixtures of singlet states.<n>We employ state-of-the-art numerical optimization methods and models to quantitatively analyze non-multiplicativity in these states for d = 3.
• Score: 3.2948779846844483
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The geometric measure of entanglement (GME) quantifies how close a multi-partite quantum state is to the set of separable states under the Hilbert-Schmidt inner product. The GME can be non-multiplicative, 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 non-multiplicativity is identified around the anti-symmetric projector state. We employ state-of-the-art numerical optimization methods and models to quantitatively analyze non-multiplicativity 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 non-multiplicative even for real separable states.

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