How entangled can a multi-party system possibly be?

TL;DR

This paper derives an upper bound for the maximum geometric measure of entanglement in multi-party quantum systems based on subsystem dimensions, providing insights into the limits of entanglement.

Contribution

It introduces a dimension-dependent upper bound for the geometric measure of entanglement in n-partite systems, enhancing understanding of entanglement limits.

Findings

Upper bound is tight in many cases

Bound depends solely on subsystem dimensions

Provides a quantitative limit for entanglement

Abstract

The geometric measure of entanglement of a pure quantum state is defined to be its distance to the space of product (seperable) states. Given an $n$-partite system composed of subsystems of dimensions $d_1,\ldots, d_n$, an upper bound for maximally allowable entanglement is derived in terms of geometric measure of entanglement. This upper bound is characterized exclusively by the dimensions $d_1,\ldots, d_n$ of composite subsystems. Numerous examples demonstrate that the upper bound appears to be reasonably tight.