# Geometric Aspects of Entanglement

## TL;DR

This paper explores quantum entanglement using geometry, introducing a new measure called entanglement distance that helps quantify entanglement in multi-qubit systems.

## Contribution

The paper introduces the entanglement distance as a geometric measure derived from the Fubini–Study metric, linking it to established entanglement measures in specific cases.

## Key findings

- Entanglement distance (ED) is derived from the Fubini–Study metric and quantifies entanglement as a geometric obstruction.
- For two-qubit pure states, ED corresponds to known measures like concurrence and entropy of entanglement.
- ED's behavior under local operations and classical communication is analyzed, showing its limitations beyond two-qubit systems.

## Abstract

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini–Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini–Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.