Entanglement in Directed Graph States

TL;DR

This paper explores entanglement in directed graph states using a geometric measure, revealing that entanglement depends only on vertex degrees and is invariant under relabeling, with implications for quantum network design.

Contribution

It introduces the Entanglement Distance as a topological measure for directed graph states, linking entanglement to vertex degree distribution and providing a geometric interpretation.

Findings

Entanglement measure depends solely on vertex degrees.

Entanglement is invariant under vertex relabeling.

Provides a geometric perspective on quantum correlations.

Abstract

We investigate a family of quantum states defined by directed graphs, where the oriented edges represent interactions between ordered qubits. As a measure of entanglement, we adopt the Entanglement Distance - a quantity derived from the Fubini - Study metric on the system's projective Hilbert space. We demonstrate that this measure is entirely determined by the vertex degree distribution and remains invariant under vertex relabeling, underscoring its topological nature. Consequently, the entanglement depends solely on the total degree of each vertex, making it insensitive to the distinction between incoming and outgoing edges. These findings offer a geometric interpretation of quantum correlations and entanglement in complex systems, with promising implications for the design and analysis of quantum networks.