Journal of Physics A Mathematical and Theoretical

SLOCC polynomial invariants of degree 6 for evennqubits

Xiangrong Li, Dafa Li

Published Mar 13, 2013• 5 citations

Abstract

We give a practical, necessary and sufficient condition for two pure n -qubit states to be stochastic local operations and classical communication (SLOCC) equivalent. We also construct the SLOCC polynomial invariants of degree 6 for even n -qubit ( n ⩾ 4) states. The polynomials are in the explicit form of 3 × 3 determinants whose entries are polynomials of degree 2 in the coefficients of the states. Further, the polynomials satisfy an SLOCC equation and therefore are invariant under SLOCC. The SLOCC equations can be used for the SLOCC classification of even n qubits.

DOI: 10.1088/1751-8113/46/13/135301

Related papers

Stochastic local operations and classical communication equations and classification of evennqubits

Xiangrong Li, Dafa Li

For any even n qubits, we establish four stochastic local operations and classical communication (SLOCC) equations and construct four SLOCC polynomials (not complete) of degree 2 n /2 , which can be exploited for the SLOCC classification (not complete) of any even n qubits. In light of the SLOCC equations, we propose several different genuine entangled states of even n qubits and show that they are inequivalent to the | GHZ ⟩, | W ⟩, or | l , n ⟩ (the symmetric Dicke states with l excitations) under SLOCC via the vanishing or not of the polynomials. The absolute values of the polynomials can be considered as entanglement measures.

arXiv:0910.4276Journal of Physics A Mathematical and TheoreticalMar 21, 20118 citations

LU Invariants and Canonical Forms and SLOCC Classification of Pure 3-Qubit States

Di Yao-Min, Ya Cao

In this paper the entanglement of pure 3-qubit states is discussed. The local unitary (LU) polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients of the canonical forms are given. Then the stochastic local operations and classical communication (SLOCC) classification of the states are discussed on the basis of the canonical forms, and the symmetric canonical form of the states without 3-tangle is discussed. Finally, we give the relation between the LU polynomial invariants and SLOCC classification.

Communications in Theoretical PhysicsApr 1, 20063 citations

Stochastic local operations and classical communication invariants via square matrix

Xin-Wei Zha, Irfan Ahmed, Da Zhang +2 more

Abstract In this paper, we raise a few corrections from the Li et al paper (2015 Phys. Rev . A 91 012302). We obtain some stochastic local operations and classical communication (SLOCC) invariants from second degree to sixth degree by defining square matrices. The relation of SLOCC invariants and the character polynomial of the square matrix are defined for three and four qubit states. This method comes as a useful resource in determining SLOCC invariants for n-qubit states.

Laser PhysicsJan 4, 20193 citations

SLOCC determinant invariants of order 2n/2for evennqubits

Xiangrong Li, Dafa Li

In this paper, we study SLOCC determinant invariants of order 2 n /2 for any even n qubits, which satisfy the SLOCC determinant equations. The determinant invariants can be constructed by a simple method and the set of all these determinant invariants is complete with respect to permutations of qubits. The SLOCC entanglement classification can be achieved via the vanishing or not of the determinant invariants. We exemplify the method for several even number of qubits, with an emphasis on six qubits.

arXiv:1106.3592Journal of Physics A Mathematical and TheoreticalFeb 2, 20127 citations

Polynomial invariants for discrimination and classification of four-qubit entanglement

Oliver Viehmann, Christopher Eltschka, Jens Siewert

The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ 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 entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.

arXiv:1101.5558Physical Review AMay 31, 201165 citations

Classification of Multipartite Entanglement of All Finite Dimensionality

Gilad Gour, Nolan R. Wallach

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.

arXiv:1304.7259Physical Review LettersAug 8, 201384 citations