© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2407.18474 · Mathematical Methods in the Applied Sciences

Optimizing Entanglement in Two‐Qubit Systems

S Luna-Hernández, Claudia Quintana, Óscar Rosas-Ortiz

Published Jan 22, 2025• 0 citations

Abstract

ABSTRACT We investigate entanglement in two‐qubit systems using a geometric representation based on the minimum of essential parameters. The latter is achieved by requiring subsystems with the same entropy, regardless of whether the state of the entire system is pure or mixed. The geometric framework is provided by a convex set that forms a right‐triangle, whose points are linked to the absolute value of just two of the coherences of the system under study. As a result, we find that optimized states of two qubits host pairs of identical populations while reducing the number of coherences involved, so they are X‐shaped. A geometric ‐measure of entanglement is introduced as the distance between the points in that represent entangled states and the closest point that defines separable states. It is shown that reproduces the results of the Hill‐Wootters concurrence , so that can be interpreted as a distance‐like entanglement measure. However, unlike , the measure also distinguishes the rank of states. The universality of the two‐qubit X‐states ensures the utility of our geometric model for studying entanglement of two‐qubit states in any configuration. To show the applicability of our approach far beyond time‐independent cases, we construct a time‐dependent two‐qubit state, traced out over the complementary components of a pure tetra‐partite system, and find that its one‐qubit states share the same entropy. The entanglement measure results bounded from above by the envelope of the minima of such entropy.

DOI: 10.1002/mma.10734

Related papers

Geometric Aspects of Entanglement

Lucio De Simone, Lorenzo Capra, Arthur Vesperini +3 more

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

quant-pharXiv:2610.03446•Entropy•Oct 5, 2026•1 citation

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

Arthur Vesperini, Ghofrane Bel-Hadj-Aissa, Lorenzo Capra +1 more

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 νμ · σμ∣ψ〉, where νμ 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.

arXiv:2307.16835•Frontiers of Physics•Apr 16, 2024•14 citations

Geometry of Quantum Entanglement

Karol Życzkowski

A geometric approach to investigation of quantum entanglement is advocated. Analyzing the space of pure states of a bipartite system we show how an entanglement measure of a gives state can be related to its distance to the closest separable state. We study geometry of the (N2 - 1)-dimensional convex body of mixed quantum states acting on an N-dimensional Hilbert space and demonstrate that it belongs to the class of sets of a constant height. The same property characterizes the set of all separable states of a two-qubit system. These results contribute to our understanding of quantum entanglement and its dynamics.

OSA Workshop on Entanglement and Quantum Decoherence•Jan 1, 2008•0 citations

Linking a distance measure of entanglement to its convex roof

Alexander Streltsov, Hermann Kampermann, Dagmar Bruß

An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.

arXiv:1006.3077•New Journal of Physics•Dec 6, 2010•106 citations

Radial Convex Geometry of Quantum States and Its Relation to Best Separable Approximation

Haonan Qiang

We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity $G(ρ)=[1-L(ρ)]/L(ρ)$ from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree $Q(ρ)=[(1-p)/p]/G(ρ)$ by comparing it with the robustness relative to the maximally mixed state. We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound $(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)$ and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely $[1-L_R]/L_R\leq[1-L_B]/L_B$. For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound $(1-p)/(2p)\leq p_0\leq Q(ρ)$. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.

quant-pharXiv:2608.20050•Aug 21, 2026•0 citations

An Extension of Entanglement Measures for Pure States

Xian Shi, Lin Chen

Abstract To quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.

arXiv:2007.13164•Annalen der Physik•Mar 17, 2021•7 citations

ABSTRACT We investigate entanglement in two‐qubit systems using a geometric representation based on the minimum of essential parameters. The latter is achieved by requiring subsystems with the same entropy, regardless of whether the state of the entire system is pure or mixed. The geometric framework is provided by a convex set that forms a right‐triangle, whose points are linked to the absolute value of just two of the coherences of the system under study. As a result, we find that optimized states of two qubits host pairs of identical populations while reducing the number of coherences involved, so they are X‐shaped. A geometric ‐measure of entanglement is introduced as the distance between the points in that represent entangled states and the closest point that defines separable states. It is shown that reproduces the results of the Hill‐Wootters concurrence , so that can be interpreted as a distance‐like entanglement measure. However, unlike , the measure also distinguishes the rank of states. The universality of the two‐qubit X‐states ensures the utility of our geometric model for studying entanglement of two‐qubit states in any configuration. To show the applicability of our approach far beyond time‐independent cases, we construct a time‐dependent two‐qubit state, traced out over the complementary components of a pure tetra‐partite system, and find that its one‐qubit states share the same entropy. The entanglement measure results bounded from above by the envelope of the minima of such entropy.

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

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 νμ · σμ∣ψ〉, where νμ 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.

A geometric approach to investigation of quantum entanglement is advocated. Analyzing the space of pure states of a bipartite system we show how an entanglement measure of a gives state can be related to its distance to the closest separable state. We study geometry of the (N2 - 1)-dimensional convex body of mixed quantum states acting on an N-dimensional Hilbert space and demonstrate that it belongs to the class of sets of a constant height. The same property characterizes the set of all separable states of a two-qubit system. These results contribute to our understanding of quantum entanglement and its dynamics.

An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.

We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity

G(ρ)=[1-L(ρ)]/L(ρ)

from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree

Q(ρ)=[(1-p)/p]/G(ρ)

by comparing it with the robustness relative to the maximally mixed state. We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound

(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)

and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely

[1-L_R]/L_R\leq[1-L_B]/L_B

. For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound

(1-p)/(2p)\leq p_0\leq Q(ρ)

. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.

Abstract To quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.