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All the separabilities of N-qubit Greenberger-Horne-Zeilinger states in white noise

Xiaoyu Chen, Li-zhen Jiang, Zhu‐An Xu

Published May 27, 2016• 0 citations
OpenAlex

Abstract

We demonstrate the necessary and sufficient criterion for an $N$ qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise when it is divided into $N-j$ parties with $N\geq 2j+1$ for arbitrary $N$ and $j$. The criterion covers more than a half of all the separabilities of $N$-qubit Greenberger-Horne-Zeilinger states in white noise. For the rest of multipartite separable problems, we present a method to obtain the sufficient conditions of separability using linear programming.

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We demonstrate the necessary and sufficient criterion for an NNN qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise when it is divided into N−jN-jN−j parties with N≥2j+1N\geq 2j+1N≥2j+1 for arbitrary NNN and jjj. The criterion covers more than a half of all the separabilities of NNN-qubit Greenberger-Horne-Zeilinger states in white noise. For the rest of multipartite separable problems, we present a method to obtain the sufficient conditions of separability using linear programming.
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.
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Abstract We characterize the separability of three qubit GHZ diagonal states in terms of entries. This enables us to check separability of GHZ diagonal states without decomposition into the sum of pure product states. In the course of discussion, we show that the necessary criterion of Gühne (2011 Entanglement criteria and full separability of multi-qubit quantum states Phys. Lett. A 375 406–10 ) for (full) separability of three qubit GHZ diagonal states is sufficient with a simpler formula. The main tool is to use entanglement witnesses which are tri-partite Choi matrices of positive bi-linear maps.
We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.
We study the noisy GHZ-W mixture (where GHZ denotes Greenberger-Horne-Zeilinger). We demonstrate some necessary but not sufficient criteria for different classes of separability of these states. It turns out that the partial transposition criterion of Peres [Phys. Rev. Lett. 77, 1413 (1996)] and the criteria of G\"uhne and Seevinck [New J. Phys. 12, 053002 (2010)] dealing with matrix elements are the strongest ones for different separability classes of this two-parameter state. As a result, we determine a set of entangled states of positive partial transpose.
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