Geometric genuine multipartite entanglement for four-qubit systems
arXiv:2212.11690 · doi:10.1016/j.physo.2024.100230
Abstract
Xie and Eberly introduced a genuine multipartite entanglement (GME) measure `concurrence fill'(\textit{Phys. Rev. Lett., \textbf{127}, 040403} (2021)) for three-party systems. It is defined as the area of a triangle whose side lengths represent squared concurrence in each bi-partition. However, it has been recently shown that concurrence fill is not monotonic under LOCC, hence not a faithful measure of entanglement. Though it is not a faithful entanglement measure, it encapsulates an elegant geometric interpretation of bipartite squared concurrences. There have been a few attempts to generalize GME measure to four-party settings and beyond. However, some of them are not faithful, and others simply lack an elegant geometric interpretation. The recent proposal from Xie et al. constructs a concurrence tetrahedron, whose volume gives the amount of GME for four-party systems; with generalization to more than four parties being the hypervolume of the simplex structure in that dimension. Here, we show by construction that to capture all aspects of multipartite entanglement, one does not need a more complex structure, and the four-party entanglement can be demonstrated using \textit{2D geometry only}. The subadditivity together with the Araki-Lieb inequality of linear entropy is used to construct a direct extension of the geometric GME to four-party systems resulting in quadrilateral geometry. Our measure can be geometrically interpreted as a combination of three quadrilaterals whose sides result from the concurrence in one-to-three bi-partition, and diagonal as concurrence in two-to-two bipartition.
arXiv comment14 pages, 3 figures
References in corpus (37)
- Quantum entanglement
- Entanglement of Formation of an Arbitrary State of Two Qubits
- Three qubits can be entangled in two inequivalent ways
- Quantum secret sharing
- Distributed Entanglement
- Quantifying Entanglement
- Universal state inversion and concurrence in arbitrary dimensions
- Four qubits can be entangled in nine different ways
- Global entanglement in multiparticle systems
- Taming multiparticle entanglement
- Measure of genuine multipartite entanglement with computable lower bounds
- How entangled can two couples get?
- Operational Families of Entanglement Classes for Symmetric -Qubit States
- Channel Capacities versus Entanglement Measures in Multiparty Quantum States
- A Triangle Governs Genuine Tripartite Entanglement
- Hierarchies of Geometric Entanglement
- Classification of multipartite entanglement in all dimensions
- Entanglement in the symmetric sector of qubits
- Computable and operationally meaningful multipartite entanglement measures
- Observable estimation of entanglement for arbitrary finite-dimensional mixed states
- Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem
- Classification of general n-qubit states under stochastic local operations and classical communication in terms of the rank of coefficient matrix
- No second law of entanglement manipulation after all
- Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product
- Geometric mean of bipartite concurrences as a genuine multipartite entanglement measure
- Genuine multipartite entanglement measure
- Dimensionally sharp inequalities for the linear entropy
- Entanglement Polygon Inequality in Qubit Systems
- Pearson Correlation Coefficient as a measure for Certifying and Quantifying High Dimensional Entanglement
- Tripartite entanglement measure under local operations and classical communication
- Coherence, Path-Predictability and I-Concurrence: A Triality
- Entanglement polygon inequality in qudit systems
- Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
- Revisiting comparison between entanglement measures for two-qubit pure states
- Comment: Inductive entanglement classification of four qubits under stochastic local operation and classical communication
- Geometric quantification of multiparty entanglement through orthogonality of vectors
- Classification and Quantification of Entanglement Through Wedge Product and Geometry