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arXiv:0910.4276 · Journal of Physics A Mathematical and Theoretical

Stochastic local operations and classical communication equations and classification of evennqubits

Xiangrong Li, Dafa Li

Published Mar 21, 2011• 8 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/15/155304

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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.
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.
We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |ℓ,n> with ℓ=1,…,[n/2] are inequivalent under SLOCC.
Recently, several schemes for the experimental creation of Dicke states were described. In this paper, we show that all the n -qubit symmetric Dicke states with l (2⩽ l ⩽( n - 2)) excitations are inequivalent to the | GHZ ⟩ state or the | W ⟩ state under SLOCC, that the even n -qubit symmetric Dicke state with n /2 excitations is inequivalent to any even n -qubit symmetric Dicke state with l ≠ n /2 excitations under SLOCC, and that all the n -qubit symmetric Dicke states with l (2⩽ l ⩽( n - 2)) excitations satisfy Coffman, Kundu and Wootters' generalized monogamy inequality C 12 2 +...+ C 1 n 2 < C 1(2... n ) 2 <1.
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.
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.
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.