Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Entanglement and separability of two-qubit states on Minkowski space with compact support

Helena Braga, Simone Souza, Salomon Sylvain Mizrahi

Published Nov 22, 2011• 3 citations
Publisher

Abstract

The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.

DOI
10.1103/physreva.84.052324

Related papers

Geometrical meaning of two-qubit entanglement and its symmetries

Helena Braga, Simone Souza, Salomon Sylvain Mizrahi

For pure and mixed two-qubit states we present an analysis based on symmetries of vectors and matrices associated to the density operator. These symmetries are revealed by doing reflection operations on the vectors and the matrices by Cartesian planes. Then we consider states whose $4\ifmmode\times\else\texttimes\fi{}4$ matrices belong to the D-$7$ manifold class and introduce a new set of parameters. This procedure allows us to establish the Peres-Horodecki separability criterion in terms of a squared distance having a Minkowski metric. Thereafter, defining a phase space in terms of those parameters, we identify two regions, a cone and elsewhere. Separable states are represented by points located within the cone or on its surface (the border), while elsewhere the points stand for the entangled states. For a system evolving in time, if its state is initially entangled, then depending on the values of its parameters and on the nature of the interactions with the environment, the trajectory in the phase space may cross the border at a finite time, going from the entangledlike to the separablelike region and vice-versa. As a particular feature, crossings in one direction stand for the phenomenon of sudden death of entanglement, while crossings in the opposite direction hint for the sudden birth of entanglement. Our method dispenses with operations on the density matrix; it is sufficient to write the expressions directly in terms of matrix entries. Some illustrative examples are worked out.

Physical Review A•Apr 16, 2010•7 citations

Minkowski structure for purity and entanglement of Gaussian bipartite states

M. C. de Oliveira, F. Nicacio, Salomon Sylvain Mizrahi

The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated with a hyperbolic space with a Minkowski metric, which is divided in two regions: separabilitylike and entanglementlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distance measures: one related to the purity and another related to amount of entanglement. The second distance allows us to define a measure for entanglement in terms of the invariant interval between the given state and its closest separable state, given in a natural manner without the requirement of a minimization procedure.

arXiv:1310.6470•Physical Review A•Nov 20, 2013•4 citations

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

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

Vector properties of entanglement in a three-qubit system

Dmitry B. Uskov, Paul M. Alsing

We suggest a dynamical vector model of entanglement in a three-qubit system based on isomorphism between su(4) and so(6) Lie algebras. This model allows one to write an evolution equation for three-qubit entanglement parameters under an arbitrary pairwise qubit coupling. Generalizing a Pl\"ucker-type description of three-qubit local invariants we introduce three pairs of real-valued three-dimensional vectors (denoted here as ${\mathbf{A}}_{R,I}$, ${\mathbf{B}}_{R,I}$, and ${\mathbf{C}}_{R,I}$). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ under local SU(2) operations is identical to SO(3) evolution of single-qubit Bloch vectors of qubits $a$, $b$, and $c$ correspondingly. At the same time, general two-qubit su(4) Hamiltonians incorporating $a\text{\ensuremath{-}}b$, $a\ensuremath{-}c$, and $b\text{\ensuremath{-}}c$ two-qubit coupling terms generate SO(6) coupling between vectors $\mathbf{A}$ and $\mathbf{B}$, $\mathbf{A}$ and $\mathbf{C}$, and $\mathbf{B}$ and $\mathbf{C}$ correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between $W$, Greenberg-Horne-Zeilinger, and biseparable states.

arXiv:2003.14390•Physical Review A•Sep 1, 2020•4 citations

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

Vladimir P. Gerdt, A. Khvedelidze, Yu. G. Palii

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices $\mathfrak{P}_ +$ . Since elements of $\mathfrak{P}_ +$ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, $\mathfrak{P}_ + \in \mathbb{R}^{15}$ . We define $\mathfrak{P}_ +$ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

Physics of Atomic Nuclei•Jun 1, 2011•6 citations
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, {s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
For pure and mixed two-qubit states we present an analysis based on symmetries of vectors and matrices associated to the density operator. These symmetries are revealed by doing reflection operations on the vectors and the matrices by Cartesian planes. Then we consider states whose $4\ifmmode\times\else\texttimes\fi{}4$ matrices belong to the D-777 manifold class and introduce a new set of parameters. This procedure allows us to establish the Peres-Horodecki separability criterion in terms of a squared distance having a Minkowski metric. Thereafter, defining a phase space in terms of those parameters, we identify two regions, a cone and elsewhere. Separable states are represented by points located within the cone or on its surface (the border), while elsewhere the points stand for the entangled states. For a system evolving in time, if its state is initially entangled, then depending on the values of its parameters and on the nature of the interactions with the environment, the trajectory in the phase space may cross the border at a finite time, going from the entangledlike to the separablelike region and vice-versa. As a particular feature, crossings in one direction stand for the phenomenon of sudden death of entanglement, while crossings in the opposite direction hint for the sudden birth of entanglement. Our method dispenses with operations on the density matrix; it is sufficient to write the expressions directly in terms of matrix entries. Some illustrative examples are worked out.
The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated with a hyperbolic space with a Minkowski metric, which is divided in two regions: separabilitylike and entanglementlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distance measures: one related to the purity and another related to amount of entanglement. The second distance allows us to define a measure for entanglement in terms of the invariant interval between the given state and its closest separable state, given in a natural manner without the requirement of a minimization procedure.
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.
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 suggest a dynamical vector model of entanglement in a three-qubit system based on isomorphism between su(4) and so(6) Lie algebras. This model allows one to write an evolution equation for three-qubit entanglement parameters under an arbitrary pairwise qubit coupling. Generalizing a Pl\"ucker-type description of three-qubit local invariants we introduce three pairs of real-valued three-dimensional vectors (denoted here as AR,I{\mathbf{A}}_{R,I}AR,I​, BR,I{\mathbf{B}}_{R,I}BR,I​, and CR,I{\mathbf{C}}_{R,I}CR,I​). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors A\mathbf{A}A, B\mathbf{B}B, and C\mathbf{C}C under local SU(2) operations is identical to SO(3) evolution of single-qubit Bloch vectors of qubits aaa, bbb, and ccc correspondingly. At the same time, general two-qubit su(4) Hamiltonians incorporating $a\text{\ensuremath{-}}b$, $a\ensuremath{-}c$, and $b\text{\ensuremath{-}}c$ two-qubit coupling terms generate SO(6) coupling between vectors A\mathbf{A}A and B\mathbf{B}B, A\mathbf{A}A and C\mathbf{C}C, and B\mathbf{B}B and C\mathbf{C}C correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors A\mathbf{A}A, B\mathbf{B}B, C\mathbf{C}C. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between WWW, Greenberg-Horne-Zeilinger, and biseparable states.
We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices P+\mathfrak{P}_ +P+​ . Since elements of P+\mathfrak{P}_ +P+​ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, P+∈R15\mathfrak{P}_ + \in \mathbb{R}^{15}P+​∈R15 . We define P+\mathfrak{P}_ +P+​ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.