Poster: The Geometric Measure of Entanglement in Networks and Exploring its Non-Multiplicativity

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 can be entangled across subsystems. However, beyond a few examples, very little is known about which states demonstrate non-multiplicativity. In this work, we first consider the GME of entangled states that arise naturally in quantum networks and show that in many but not all cases the GME is multiplicative among the edges in the network. We then focus on symmetric families of two qutrit states and numerically investigate when the GME of the states is non-multiplicative. We derive local optima in terms of the parameters that define our states and numerically verify that the lower bound of the global optima leads to a violation. We show that separable states violate the multiplicative property when we only allow for real-vector optimization, but no separable states violate over complex-vector optimization. We show that this is not true for all families of states.