The concurrence vectors are proposed by employing the fundamental representation of $A_n$ Lie algebra, which provides a clear criterion to evaluate the entanglement of bipartite system of arbitrary dimension. A state is separable if the norm of its concurrence vector vanish. The sign situation of the nonzero components of concurrence vectors of entangled bases presents a simple criterion for whether the whole Hilbert subspace spanned by those bases is of entangled, or there exists entanglement edge in the subspace. Concurrence surfaces are observed for several concrete examples.The concurrence vectors are proposed by employing the fundamental representation of $A_n$ Lie algebra, which provides a clear criterion to evaluate the entanglement of bipartite system of arbitrary dimension. A state is separable if the norm of its concurrence vector vanish. The sign situation of the nonzero components of concurrence vectors of entangled bases presents a simple criterion for whether the whole Hilbert subspace spanned by those bases is of entangled, or there exists entanglement edge in the subspace. Concurrence surfaces are observed for several concrete examples.

Aug 26, 200312 citations

Geometric interpretation of the triality relation in bipartite quantum systems

Pratidhwani Swain, Ramita Sarkar, Sukanta K. Tripathy +1 more

Abstract We investigate quantum correlations and coherence in bipartite systems using the universal triality relation that connects coherence, predictability, and entanglement. The geometric structure underlying entanglement in bipartite two-qubit and two-qutrit systems is analyzed using wedge products of post-measurement vectors. For two-qubit states, concurrence emerges naturally as the area spanned by the post-measurement vectors, while in qutrit systems the generalized I-concurrence is associated with higher-dimensional geometric structures formed by multiple wedge products. We further analyze the relationship between wedge products, eigenvalues, and eigenvectors of correlated systems and discuss the universality of the triality relation in arbitrary dimensions. Our results provide a geometric interpretation of entanglement in d -dimensional cases and reveal a hierarchical structure connecting low- and high-dimensional quantum systems.Abstract We investigate quantum correlations and coherence in bipartite systems using the universal triality relation that connects coherence, predictability, and entanglement. The geometric structure underlying entanglement in bipartite two-qubit and two-qutrit systems is analyzed using wedge products of post-measurement vectors. For two-qubit states, concurrence emerges naturally as the area spanned by the post-measurement vectors, while in qutrit systems the generalized I-concurrence is associated with higher-dimensional geometric structures formed by multiple wedge products. We further analyze the relationship between wedge products, eigenvalues, and eigenvectors of correlated systems and discuss the universality of the triality relation in arbitrary dimensions. Our results provide a geometric interpretation of entanglement in d -dimensional cases and reveal a hierarchical structure connecting low- and high-dimensional quantum systems.

Physica ScriptaSep 7, 20260 citations

Advances in high-dimensional quantum entanglement

Manuel Erhard, Mario Krenn, Anton Zeilinger

arXiv:1911.10006Nature Reviews PhysicsJun 25, 2020650 citations

Concurrence vectors in arbitrary multipartite quantum systems

Seyed Javad Akhtarshenas

For a given pure state of a multipartite system, the concurrence vector is defined by employing the defining representation of generators of the corresponding rotation groups. The norm of a concurrence vector is considered as a measure of entanglement. For the multipartite pure state, the concurrence vector is regarded as the direct sum of concurrence subvectors in the sense that each subvector is associated with a pair of particles. It is proposed to use the norm of each subvector as the contribution of the corresponding pair in entanglement of the system.For a given pure state of a multipartite system, the concurrence vector is defined by employing the defining representation of generators of the corresponding rotation groups. The norm of a concurrence vector is considered as a measure of entanglement. For the multipartite pure state, the concurrence vector is regarded as the direct sum of concurrence subvectors in the sense that each subvector is associated with a pair of particles. It is proposed to use the norm of each subvector as the contribution of the corresponding pair in entanglement of the system.

arXiv:quant-ph/0311166Journal of Physics A Mathematical and GeneralJul 13, 200584 citations

Swapped entanglement in high-dimensional quantum systems

S. M. Zangi, Chitra Shukla, Khalid Naseer +1 more

arXiv:2508.00634Quantum Information ProcessingJul 23, 20261 citation

The concurrence vectors are proposed by employing the fundamental representation of

A_n

Lie algebra, which provides a clear criterion to evaluate the entanglement of bipartite system of arbitrary dimension. A state is separable if the norm of its concurrence vector vanish. The sign situation of the nonzero components of concurrence vectors of entangled bases presents a simple criterion for whether the whole Hilbert subspace spanned by those bases is of entangled, or there exists entanglement edge in the subspace. Concurrence surfaces are observed for several concrete examples.