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arXiv:0805.2472 · Europhysics Letters (EPL)

Stochastic local operations and classical communication properties of the n-qubit symmetric Dicke states

Dafa Li, Xinxin Li, Hongtao Huang, Xiangrong Li

Published Jul 1, 2009• 11 citations
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Abstract

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.

DOI
10.1209/0295-5075/87/20006

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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.
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 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.
We show that all multipartite pure states that are stochastic local operation and classical communication (SLOCC) equivalent to the NNN-qubit WWW state can be uniquely determined (among arbitrary states) from their bipartite marginals. We also prove that only $(N\ensuremath{-}1)$ of the bipartite marginals are sufficient and that this is also the optimal number. Thus, contrary to the Greenberger-Horne-Zeilinger (GHZ) class, WWW-type states preserve their reducibility under SLOCC. We also study the optimal reducibility of some larger classes of states. The generic Dicke states $|{\text{GD}}_{N}^{\ensuremath{\ell}}\ensuremath{\rangle}$ are shown to be optimally determined by their ($\ensuremath{\ell}+1$)-partite marginals. The class of ``GGG'' states (superposition of WWW and W‾\overline{W}W) are shown to be optimally determined by just two $(N\ensuremath{-}2)$-partite marginals.
We study the interconversion of multipartite symmetric NNN-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric NNN-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.
We introduce a notion of the entanglement transformation rate to characterize the asymptotic comparability of two multipartite pure entangled states under stochastic local operations and classical communication (SLOCC). For two well known SLOCC inequivalent three-qubit states |GHZ⟩=(1/2)(|000⟩+|111⟩) and |W⟩=(1/3)(|100⟩+|010⟩+|001⟩), we show that the entanglement transformation rate from |GHZ⟩ to |W⟩ is exactly 1. That means that we can obtain one copy of the W state from one copy of the Greenberg-Horne-Zeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an N-partite GHZ state to a class of Dicke states, and prove the tightness of this bound for some special cases which naturally generalize the |W⟩ state. A new lower bound on the tensor rank of the matrix permanent is also obtained by evaluating the tensor rank of Dicke states.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.