Classical Tensors and Quantum Entanglement I: Pure States

TL;DR

This paper explores the geometric tensor structures of quantum state spaces, focusing on how they encode properties of pure separable and entangled states using Riemannian and symplectic geometry.

Contribution

It introduces a geometric framework for analyzing pure quantum states, linking tensor fields to state properties and relating to existing criteria for entanglement.

Findings

Geometric tensors encode properties of separable and entangled states.

Pull-back tensor fields relate classical manifolds to quantum state spaces.

Connections to existing entanglement criteria are explicitly discussed.

Abstract

The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.