Unveiling the geometric meaning of quantum entanglement: Discrete 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 Phys. Rev. A 101, 042129 (2020). 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 ${\mathit{v}}^{\mu }\cdot {\mathit{\sigma }}^{\mu }|\psi ⟩$, where ${\mathit{v}}^{\mu }$ are unit vectors and $\mu$ runs on the number of parties. Within the proposed geometric approach, we derive a general method to determine when two states are not the same state up to the action of local unitary operators. Furthermore, we prove that the entanglement distance, along with its convex roof expansion to mixed states, fulfils 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 does not 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 application for the case of a system with continuous variables, we have considered a system of two coupled Glauber coherent states.

Graphical abstract

Keywords

entanglements / quantum information / entanglement measure

References

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

We acknowledge the support from the Research Support Plan 2022 − Call for applications for funding allocation to research projects curiosity driven (F CUR) − Project “Entanglement Protection of Qubits’ Dynamics in a Cavity” – EPQDC and the support by the Italian National Group of Mathematical Physics (GNFM-INdAM). R. F. and A. V. would like to acknowledge INFN Pisa for the financial support to this activity.

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