Unveiling the geometric meaning of quantum entanglement: discrete and continuous variable systems

TL;DR

This paper explores the geometric structure of quantum state manifolds, linking entanglement to a Riemannian metric, and introduces the entanglement distance as a measure with proven properties, applicable to both discrete and continuous variable systems.

Contribution

It derives the Fubini-Study metric for multi-qubit systems, introduces the entanglement distance as a valid entanglement measure, and extends the geometric approach to continuous variable quantum systems.

Findings

Entanglement distance equals twice the squared concurrence for two-qubit pure states.

The entanglement distance fulfills all criteria for an entanglement measure.

The geometric approach applies to various entangled state families and continuous variable systems.

Abstract

We show that the manifold of quantum states is endowed with a rich and nontrivial geometric structure. We derive the Fubini-Study metric of the projective Hilbert space of a multi-qubit quantum system, endowing it with a Riemannian metric structure, and investigate its deep link with the entanglement of the states of this space. As a measure, we adopt the Entanglement Distance E preliminary proposed in [1]. Our analysis shows that entanglement has a geometric interpretation: E(|psi>) is the minimum value of the sum of the squared distances between |psi> and its conjugate states, namely the states v^mu . sigma^mu |psi>, where v^mu are unit vectors and mu runs on the number of parties. We derive a general method to determine when two states are not the same state up to the action of local unitary operators. We prove that the entanglement distance, along with its convex roof expansion to mixed states, fulfills the three conditions required for an entanglement measure: that is i) E(|psi>) =0 iff |psi> is fully separable; ii) E is invariant under local unitary transformations; iii) E doesn't increase under local operation and classical communications. Two different proofs are provided for this latter property. We also show that in the case of two qubits pure states, the entanglement distance for a state |psi> coincides with two times the square of the concurrence of this state. We propose a generalization of the entanglement distance to continuous variable systems. Finally, we apply the proposed geometric approach to the study of the entanglement magnitude and the equivalence classes properties, of three families of states linked to the Greenberger-Horne-Zeilinger states, the Briegel Raussendorf states and the W states. As an example of an application for the case of a system with continuous variables, we have considered a system of two coupled Glauber coherent states.