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arXiv:1105.0587 · Physical Review Letters

Greenberger-Horne-Zeilinger States and Few-Body Hamiltonians

Paolo Facchi A, Giuseppe Florio, Saverio Pascazio, Francesco Vincenzo Pepe

Published Dec 19, 2011• 20 citations
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Abstract

The generation of Greenberger-Horne-Zeilinger (GHZ) states is a crucial problem in quantum information. We derive general conditions for obtaining GHZ states as eigenstates of a Hamiltonian. We find that a necessary condition for an n-qubit GHZ state to be a nondegenerate eigenstate of a Hamiltonian is the presence of m-qubit couplings with m≥[(n+1)/2]. Moreover, we introduce a Hamiltonian with a GHZ eigenstate and derive sufficient conditions for the removal of the degeneracy.

DOI
10.1103/physrevlett.107.260502

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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.
A Multipartite entangled state has many different kinds of entanglements specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger–Horne–Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For an [Formula: see text] qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into [Formula: see text] parties with [Formula: see text] for arbitrary [Formula: see text] and [Formula: see text]. The criterion covers more than a half of all kinds of partial entanglements for [Formula: see text]-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider [Formula: see text] qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.
The [Formula: see text]-qubit Greenberger–Horne–Zeilinger (GHZ) states are the maximally entangled states of [Formula: see text] qubits, which have had many important applications in quantum information processing, such as quantum key distribution and quantum secret sharing. Thus how to distinguish the GHZ states from other quantum states becomes a significant problem. In this work, by presenting a family of the generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, we show that the [Formula: see text]-qubit GHZ states can be indeed identified by the maximal violations of the generalized CHSH inequality under some specific measurement settings. The generalized CHSH inequality is simple and contains only four correlation functions for any [Formula: see text]-qubit system, thus has the merit of facilitating experimental verification. Furthermore, we present a quantum phenomenon of robust violations of the generalized CHSH inequality in which the maximal violation of Bell’s inequality can be robust under some specific noises adding to the [Formula: see text]-qubit GHZ states.