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Physical Review A

Determination of Dicke states equivalent under stochastic local operations and classical communication

Xia Wu, Ying‐Hui Yang, Qiaoyan Wen, Su‐Juan Qin, Fei Gao

Published Nov 30, 2015• 3 citations
Publisher

Abstract

It has been proven by Rana et al. that the $n$-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 $n$-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 $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.

DOI
10.1103/physreva.92.052338

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It has been proven by Rana et al. that the nnn-qubit generic Dicke states ∣GD(n,\ensuremathℓ)\ensuremath⟩|\mathrm{GD}(n,\ensuremath{\ell})\ensuremath{\rangle}∣GD(n,\ensuremathℓ)\ensuremath⟩ are uniquely determined, among arbitrary states, by their (\ensuremathℓ+1)(\ensuremath{\ell}+1)(\ensuremathℓ+1)-partite reduced density matrices, and (n\ensuremath−1\ensuremathℓ)(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})(n\ensuremath−1\ensuremathℓ​) 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 (\ensuremathℓ+1)(\ensuremath{\ell}+1)(\ensuremathℓ+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,\ensuremathℓ)\ensuremath⟩|D(n,\ensuremath{\ell})\ensuremath{\rangle}∣D(n,\ensuremathℓ)\ensuremath⟩. Moreover, we find two kinds of subsets of the (\ensuremathℓ+1)(\ensuremath{\ell}+1)(\ensuremathℓ+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 (n\ensuremath−1\ensuremathℓ)(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})(n\ensuremath−1\ensuremathℓ​).
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.
The determination of many special types of quantum states has been studied thoroughly, such as the generalized $|\text{GHZ}\ensuremath{\rangle}$ states, stochastic local operations and classical communication equivalent $|W\ensuremath{\rangle}$ states, and Dicke states. In this paper, we study another special entanglement state, that is, stabilizer states, which play an important role in quantum error correcting codes, multipartite purification, and so on. We show that all nnn-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of nnn independent generators of the corresponding stabilizer formalisms.
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.
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
Only nnn-qubit Greenberger-Horne-Zeilinger states contain nnn-partite information
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
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.