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Bulletin of the American Physical Society

Only $n$-qubit Greenberger-Horne-Zeilinger states contain $n$-partite information

Scott N. Walck, David W. Lyons

Published May 21, 2009• 1 citation
OpenAlex

Abstract

The generalized $n$-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of $n$ 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 $n$-party level.

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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−1n\ensuremath{-}1n\ensuremath−1 qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
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.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
We investigate local contents in NNN-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states. We suggest a decomposition for correlations in the GHZ states into a nonlocal and fully local part, and find a lower and upper bound on the local content. Our lower bound reproduces the previous result for N=2N=2N=2 [V. Scarani, Phys. Rev. A 77, 042112 (2008)] and decreases rapidly with NNN.
We show that not all four-party pure states are Greenberger-Horne-Zeilinger (GHZ) reducible (i.e., can be generated reversibly from a combination of two-, three-, and four-party maximally entangled states by local quantum operations and classical communication asymptotically). We also present some properties of the relative entropy of entanglement for those three-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.