Polynomial invariants for discrimination and classification of four-qubit entanglement

Abstract

It is well known that the number of entanglement classes in SLOCC (stochastic local operations and classical communication) classifications increases with the number of qubits and is already infinite for four qubits. Bearing in mind the rapid evolution of experimental technology, criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore in the focus of intense theoretical research. In this article we develop a general criterion for the discrimination of pure N-partite entangled states in terms of polynomial SL(d,C) invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which families are identified through 'tangle patterns', thus bringing together qualitative and quantitative description of entanglement.

9 pages, 3 tables

References in corpus (22)

• Three qubits can be entangled in two inequivalent ways
• Entanglement of a Pair of Quantum Bits
• Distributed Entanglement
• Quantifying Entanglement
• Entanglement monotones
• Four qubits can be entangled in nine different ways
• A potential multipartide entanglement measure
• Normal forms and entanglement measures for multipartite quantum states
• The Schmidt Measure as a Tool for Quantifying Multi-Particle Entanglement
• Constructing N-qubit entanglement monotones from anti-linear operators
• The polynomial invariants of four qubits
• Computing Local Invariants of Qubit Systems
• Operational Families of Entanglement Classes for Symmetric $N$-Qubit States
• Local unitary equivalence of multipartite pure states
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Four-qubit entanglement from string theory
• On polynomial invariants of several qubits
• Measuring Polynomial Invariants of Multi-Party Quantum States
• Multipartite entanglement in 2 x 2 x n quantum systems
• Evolution and Symmetry of Multipartite Entanglement
• Generation of bipartite spin entanglement via spin-independent scattering
• The invariant-comb approach and its relation to the balancedness of multipartite entangled states

Cited by in corpus (47)

• Quantifying entanglement resources
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms
• Convex geometry of quantum resource quantification
• Classification of multipartite entanglement in all dimensions
• Classification of general n-qubit states under stochastic local operations and classical communication in terms of the rank of coefficient matrix
• Multipartite Entanglement Measure and Complete Monogamy Relation
• Genuine multipartite entanglement measure
• Multipartite-entanglement monotones and polynomial invariants
• Classification of Entanglement in Symmetric States
• Partial transpose criteria for symmetric states
• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• The n-tangle of odd n qubits
• Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states
• Deterministic generation of arbitrary symmetric states and entanglement classes
• Rescaling multipartite entanglement measures for mixed states
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• SLOCC Invariants for Multipartite Mixed States
• Operational Entanglement Families of Symmetric Mixed N-Qubit States
• Entanglement classification of 2 X 2 X 2 X d quantum systems via the ranks of the multiple coefficient matrices
• Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states
• Classification scheme of pure multipartite states based on topological phases
• Classifying Quantum Entanglement through Topological Links
• A brief introduction to multipartite entanglement
• Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix
• Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices
• Complete monogamy of the multipartite quantum mutual information
• Polynomial invariants of degree 4 for even-$n$ qubits and their applications in entanglement classification
• Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices
• Unitary Invariants and Classification of Four-Qubit States via Negativity Fonts
• Criterion for SLOCC Equivalence of Multipartite Quantum States
• Generalized W-state of four qubits with exclusively threetangle
• SLOCC determinant invariants of order 2^{n/2} for even n qubits
• A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
• Classification of multipartite systems featuring only $|W⟩$ and $|GHZ⟩$ genuine entangled states
• Complete Genuine Multipartite Entanglement Monotone
• Negativity Fonts, multiqubit invariants and Four qubit Maximally Entangled States
• Classification witness operator for the classification of different subclasses of three-qubit GHZ class
• Sequential generation of Polynomial Invariants and N-body non-local correlations
• Local unitary symmetries and entanglement invariants
• Freudenthal ranks: GHZ vs. W
• A type of localized quadripartite entanglement
• Entanglement Classification via Single Entanglement Measure
• A Note on State Decomposition Independent Local Invariants
• Multiphoton, multimode state classification for nonlinear optical circuits
• Absolutely maximally entangled pure states of multipartite quantum systems
• Entanglement classification via integer partitions