Entanglement Constraints on States Locally Connected to the Greenberger-Horne-Zeilinger State

TL;DR

This paper derives a new inequality constraining three-qubit entanglement, strengthening previous bounds, and explores entanglement sharing in GHZ states using algebraic geometry, advancing understanding of quantum entanglement limits.

Contribution

It introduces a necessary and sufficient inequality on 3-qubit tangles and unifies entanglement constraints through algebraic geometry, extending the concept of strong monogamy.

Findings

Derived a stronger achievability inequality for 3-qubit tangles.

Established a precise account of entanglement sharing in GHZ states.

Connected entanglement constraints with algebraic geometry principles.

Abstract

The multi-qubit GHZ state possesses tangles with elegant transformation properties under stochastic local operations and classical communication. Since almost all pure 3-qubit states are connected to the GHZ state via SLOCC, we derive a necessary and sufficient achievability inequality on arbitrary 3-qubit tangles, which is a strictly stronger constraint than both the monogamy inequality and the marginal eigenvalue inequality. We then show that entanglement shared with any single party in the n-qubit GHZ SLOCC equivalence class is precisely accounted for by the sum of its k-tangles, recently coined the strong monogamy equality, acknowledging competing but agreeing definitions of the k-tangle on this class, one of which is then computable for arbitrary mixed states. Strong monogamy is known to not hold arbitrarily, and so we introduce a unifying outlook on entanglement constraints in light of basic real algebraic geometry.