Geometric quantification of multiparty entanglement through orthogonality of vectors

TL;DR

This paper explores geometric conditions for maximizing multiparty entanglement in quantum systems using vector orthogonality, deriving conditions for maximally entangled states like Bell and GHZ states, and analyzing the trade-offs between local and global properties.

Contribution

It introduces a geometric framework based on vector orthogonality to characterize and identify maximally entangled states in multiparty quantum systems, extending to qudits.

Findings

Maximally entangled states correspond to specific geometric conditions of vectors.

Derived conditions for maximally entangled two-qudit and multi-qubit states.

Analyzed the trade-off between local predictability and global entanglement.

Abstract

The wedge product of vectors has been shown to yield the generalised entanglement measure I-concurrence, wherein the separability of the multiparty qubit system arises from the parallelism of vectors in the underlying Hilbert space of the subsystems. Here, we demonstrate the geometrical conditions of the post-measurement vectors which maximize the entanglement corresponding to the bi-partitions and can yield non-identical set of maximally entangled states. The Bell states for the two qubit case, GHZ and GHZ like states with superposition of four constituents for three qubits, naturally arise as the maximally entangled states. The geometric conditions for maximally entangled two qudit systems are derived, leading to the generalised Bell states, where the reduced density matrices are maximally mixed. We further show that the reduced density matrix for an arbitrary finite dimensional subsystem of a general qudit state can be constructed from the overlap of the post-measurement vectors. Using this approach, we discuss the trade-off between the local properties namely predictability and coherence with the global property, entanglement for the non-maximally entangled two qubit state.