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arXiv:1110.4597 · Physical Review A

Optimal reducibility of all W states equivalent under stochastic local operations and classical communication

Swapan Rana, Preeti Parashar

Published Nov 29, 2011• 10 citations
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Abstract

We show that all multipartite pure states that are stochastic local operation and classical communication (SLOCC) equivalent to the $N$-qubit $W$ 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, $W$-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 ``$G$'' states (superposition of $W$ and $\overline{W}$) are shown to be optimally determined by just two $(N\ensuremath{-}2)$-partite marginals.

DOI
10.1103/physreva.84.052331

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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)(N\ensuremath{-}1)(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 ∣GDN\ensuremathℓ\ensuremath⟩|{\text{GD}}_{N}^{\ensuremath{\ell}}\ensuremath{\rangle}∣GDN\ensuremathℓ​\ensuremath⟩ are shown to be optimally determined by their (\ensuremathℓ+1\ensuremath{\ell}+1\ensuremathℓ+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)(N\ensuremath{-}2)(N\ensuremath−2)-partite marginals.
It has been proven by Rana et al. that the nnn-qubit generic Dicke states $|\mathrm{GD}(n,\ensuremath{\ell})\ensuremath{\rangle}$ are uniquely determined, among arbitrary states, by their $(\ensuremath{\ell}+1)$-partite reduced density matrices, and $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$ number of them which have one party common to all are sufficient. We show that among arbitrary states, Dicke-type states are also uniquely determined by their $(\ensuremath{\ell}+1)$-partite reduced density matrices. Here, by Dicke-type states, we mean all multiqubit pure states that are stochastic local operations and classical communication equivalent to the nnn-qubit standard Dicke state $|D(n,\ensuremath{\ell})\ensuremath{\rangle}$. Moreover, we find two kinds of subsets of the $(\ensuremath{\ell}+1)$-partite reduced density matrices set, both of which can determine the Dicke-type states. More importantly, the number of elements in these two subsets is much smaller than $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.
It has been known that, among pure states, NNN-qubit WWW states cannot be uniquely determined by their arbitrary $(N\ensuremath{-}1)$ bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, $(N\ensuremath{-}1)$ bipartite reduced density matrices that the pairs of qubits constitute a star graph or a line graph can uniquely determine stochastic local operations and classical communication (SLOCC) equivalent WWW states, and we generalize this conclusion into tree graph. In this paper, we show that all SLOCC equivalent WWW states can be uniquely determined (among pure, mixed states) by their $(N\ensuremath{-}1)$ bipartite reduced density matrices, if the $(N\ensuremath{-}1)$ pairs of qubits constitute a tree graph on NNN vertices, where each pair of qubits represents an edge.
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.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
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 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.