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

doi:10.1007/s11467-024-1403-x

Front. Phys.

2024, Vol. 19

Issue (5) : 51204 https://doi.org/10.1007/s11467-024-1403-x

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

Arthur Vesperini^1,^2,³, Ghofrane Bel-Hadj-Aissa^1,^2,³, Lorenzo Capra¹, Roberto Franzosi^1,^2,³(

)

¹. DSFTA, University of Siena, Via Roma 56, 53100 Siena, Italy
². QSTAR & CNR - Istituto Nazionale di Ottica, Largo Enrico Fermi 2, I-50125 Firenze, Italy
³. INFN Sezione di Perugia, I-06123 Perugia, Italy

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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: $E (| ψ ⟩)$ is the minimum value of the sum of the squared distances between $| ψ ⟩$ and its conjugate states, namely the states $v μ ⋅ σ μ | ψ ⟩$ , where $v μ$ are unit vectors and $μ$ 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) $E (| ψ ⟩) = 0$ iff $| ψ ⟩$ 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 $| ψ ⟩$ 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.

Keywords entanglements quantum information entanglement measure

Corresponding Author(s): Roberto Franzosi

About author: Li Liu and Yanqing Liu contributed equally to this work.

Issue Date: 15 April 2024

Cite this article:

Arthur Vesperini,Ghofrane Bel-Hadj-Aissa,Lorenzo Capra, et al. Unveiling the geometric meaning of quantum entanglement: Discrete and continuous variable systems[J]. Front. Phys. , 2024, 19(5): 51204.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-024-1403-x
https://academic.hep.com.cn/fop/EN/Y2024/V19/I5/51204

Fig.1 This figure reports the three-dimensional plot of the ED

E (| W, α ? 3) / 3

as a function of

2 θ 1 / π

and

2 θ 2 / π

for the states (66), in the case

M = 3

Fig.2 The scheme in the figure represents the topological structure of the equivalence classes for some of the states for each of the three families. A point along one of the black lines represents a state of each family, from left to right these are the

α

W

states, the

?

-BRS and the

θ

-GHZLS. The magenta cloudlet represents the equivalence class to which belong the (fully separable) states. The latter are, in the case of the

α

-W states, the states

| W, α ? 2

with

α = (1, 0), (0, 1)

, in the case of

?

-BRS, the states

| r, ? ? 2

with

? = 0, 2 π

and, in the case of the

θ

-GBZLS the states

| G H Z, θ ? 2

, where

θ = 0, π / 2

. In the case of

M = 2

qubits, the states of the three families with the higher degree of entanglement, that is

| W, (1 / 2, 1 / 2) ? 2

| r, π ? 2

and

| G H Z, π / 4 ? 2

, belong to the same equivalence class, figured with a red cloudlet.

Fig.3 As in Fig.2, we report here a scheme that represents the topological structure of the equivalence classes for some of the states of the three families: the

α

W

states, the

?

-BRS and the

θ

-GHZLS. In this case, we consider three qubits states. The magenta cloudlet represents the equivalence class to which belong the (fully separable) states for the three families:

| W, α ? 3

with

α = (1, 0, 0), (0, 1, 0), (0, 0, 1)

| r, ? ? 3

with

? = 0, 2 π

and,

| G H Z, θ ? 3

, with

θ = 0, π / 2

. In the case of

M = 3

qubits, the equivalence classes of the states of the three families with the higher degree of entanglement do not coincide. In fact, the class

[| W, (1 / 3, 1 / 3, 1 / 3) ? 3]

is dijointed from the class

[| r, π ? 3] = [| G H Z, π / 4 ? 3]

, as figured with the two red cloudlets.

Fig.4 We report here a scheme analogous to those of Fig.2 and Fig.3. In this case, we consider

M = 4

qubits states. The magenta cloudlet represents the equivalence class to which belong the (fully separable) states of the three families:

| W, α ? 4

with

α = (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)

| r, ? ? 4

with

? = 0, 2 π

and,

| G H Z, θ ? 4

, with

θ = 0, π / 2

. In this case, the equivalence classes of the states of the three families with the higher degree of entanglement are all disjointed. In fact, the class

[| W, (1 / 4, 1 / 4, 1 / 4, 1 / 4) ? 4] ≠ [| r, π ? 4] ≠ [| G H Z, π / 4 ? 4]

, as figured with the three dijointed red cloudlets.

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