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Physical Review A

Determination of W states equivalent under stochastic local operations and classical communication by their bipartite reduced density matrices with tree form

Xia Wu, Guojing Tian, Wei Huang, Qiaoyan Wen, Su‐Juan Qin, Fei Gao

Published Jul 14, 2014• 7 citations
Publisher

Abstract

It has been known that, among pure states, $N$-qubit $W$ 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 $W$ states, and we generalize this conclusion into tree graph. In this paper, we show that all SLOCC equivalent $W$ 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 $N$ vertices, where each pair of qubits represents an edge.

DOI
10.1103/physreva.90.012317

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It has been known that, among pure states, NNN-qubit WWW states cannot be uniquely determined by their arbitrary (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1) bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, (N\ensuremath−1)(N\ensuremath{-}1)(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)(N\ensuremath{-}1)(N\ensuremath−1) bipartite reduced density matrices, if the (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1) pairs of qubits constitute a tree graph on NNN vertices, where each pair of qubits represents an edge.
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.
A set of necessary and sufficient conditions are derived for the equivalence of an arbitrary pure state and a graph state on n qubits under stochastic local operations and classical communication (SLOCC), using the stabilizer formalism. Because all stabilizer states are equivalent to graph states by local unitary transformations, these conditions constitute a classical algorithm for the determination of SLOCC-equivalence of pure states and stabilizer states. This algorithm provides a distinct advantage over the direct solution of the SLOCC-equivalence condition |ψ〉 = S|g〉 for an unknown invertible local operator S, as it usually allows for easy detection of states that are not SLOCC-equivalent to graph states.
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 $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.
Abstract We study the stochastic local operation and classical communication (SLOCC) equivalence for arbitrary dimensional multipartite quantum states. For multipartite pure states, we present a necessary and sufficient criterion in terms of their coefficient matrices. This condition can be used to classify some SLOCC equivalent quantum states with coefficient matrices having the same rank. For multipartite mixed state, we provide a necessary and sufficient condition by means of the realignment of matrix. Some detailed examples are given to identify the SLOCC equivalence of multipartite quantum states.
We prove that the most general WWW class of NNN-qubit states are uniquely determined among arbitrary states (pure or mixed) by just their bipartite reduced density matrices. Moreover, if we consider only pure states, then $(N\ensuremath{-}1)$ of them are shown to be sufficient.
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.