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arXiv:quant-ph/0701032 · Physical Review A

Classification of four-qubit states by means of a stochastic local operation and the classical communication invariant and semi-invariants

Dafa Li, Xiangrong Li, Hongtao Huang, Xinxin Li

Published Nov 15, 2007• 36 citations
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Abstract

We show there are at least 28 distinct true stochastic local operations and classical communication (SLOCC) entanglement classes for four qubits by means of SLOCC invariant and semi-invariants and derive the number of degenerate SLOCC classes for $n$ qubits.

DOI
10.1103/physreva.76.052311

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We show there are at least 28 distinct true stochastic local operations and classical communication (SLOCC) entanglement classes for four qubits by means of SLOCC invariant and semi-invariants and derive the number of degenerate SLOCC classes for nnn qubits.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
We find an invariant for nnn qubits and propose residual entanglement for nnn qubits by means of the invariant. Thus, we establish a relation between stochastic local operations and classical communication (SLOCC) entanglement and residual entanglement. Invariant and residual entanglement can be used for SLOCC entanglement classification of nnn qubits.
Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors [Phys. Rev. A 74, 052336 (2006)], we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.
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.
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.