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arXiv:0809.4394 · Physical Review A

N -qubit W states are determined by their bipartite marginals

Preeti Parashar, Swapan Rana

Published Jul 17, 2009• 36 citations
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Abstract

We prove that the most general $W$ class of $N$-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.

DOI
10.1103/physreva.80.012319

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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)(N\ensuremath{-}1)(N\ensuremath−1) of them are shown to be sufficient.
Identifying whether a mixed quantum state is uniquely determined among all states (UDA) by its local marginals is a basic problem in quantum information theory. We establish necessary and sufficient conditions under which several classes of multipartite mixed states are UDA by their kkk-partite marginals. We also prove structural properties based on ranges and marginals, and formulate a recursive procedure for the determination of UDA states. We show that sufficiently high rank rules out unique determination from fixed-order marginals, implying that almost all multipartite mixed states are not UDA from such marginals. Finally, we completely characterize the additivity of bipartite UDA states, three-qubit and several families of nnn-qubit product UDA states. These results clarify the boundary between UDA and non-UDA mixed states and provide a framework for local-marginal reconstruction and related certification tasks.
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.
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.
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 is known that the nnn-qubit WWW-type state is determined by its bipartite reduced density matrices. In this paper, we show that the multipartite WWW-type state is uniquely determined by its reduced density matrices among WWW-type states in the sense of local unitary equivalence.
Consider a qubit-qutrit (2×32 \times 32×3) composite state space. Let $C(\{1}{2}I_2, \{1}{3}I_3)$ be a convex set of all possible states of composite system whose marginals are given by $\{1}{2}I_2$ and $\{1}{3}I_3$ in two and three dimensional spaces respectively. We prove that there exists no pure state in $C(\{1}{2}I_2, \{1}{3}I_3)$. Further we generalize this result to an arbitrary m×nm \times nm×n bipartite systems. We prove that for m<nm < nm<n, no pure state exists in the convex set C(ρA,ρB)C(ρ_A,ρ_B)C(ρA​,ρB​), for an arbitrary ρAρ_AρA​ and rank of ρB>mρ_B >mρB​>m.