Geometry of quantum state space and quantum correlations

TL;DR

This paper explores the geometric structure of quantum state space, specifically a Riemannian metric on 2x2 density matrices, and relates it to quantum entanglement measures, suggesting a geometric perspective on quantum correlations.

Contribution

It identifies a specific Riemannian metric on 2x2 quantum states and links it to entanglement negativity, providing a geometric interpretation of quantum correlations.

Findings

Negativity of entangled states is proportional to the square root of a specific Riemannian metric.

The geometric quantity directly relates to a measure of entanglement.

The work suggests a geometric approach to understanding quantum correlations.

Abstract

Quantum state space is endowed with a metric structure and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by 2x2 density matrices, we determine a particular Riemannian metric for a state \r{ho} and show that if \r{ho} gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equals to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach.