From Geometric Quantum Mechanics to Quantum Information

TL;DR

This paper explores the geometric structure of quantum mechanics, focusing on the Fubini-Study metric and its relation to quantum information measures, providing a foundation for identifying entanglement invariants.

Contribution

It introduces a geometric framework for quantum mechanics using the Fubini-Study metric and develops an algorithmic method to find entanglement monotone candidates.

Findings

Established a link between Fubini-Study and quantum Fisher information metrics

Proposed an algorithmic approach for entanglement monotone identification

Analyzed the geometric properties of quantum state orbits

Abstract

We consider the geometrization of quantum mechanics. We then focus on the pull-back of the Fubini-Study metric tensor field from the projective Hibert space to the orbits of the local unitary groups. An inner product on these tensor fields allows us to obtain functions which are invariant under the considered local unitary groups. This procedure paves the way to an algorithmic approach to the identification of entanglement monotone candidates. Finally, a link between the Fubini-Study metric and a quantum version of the Fisher information metric is discussed.