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Chinese Physics B

Decay of N -qubit GHZ states in Pauli channels

Xiaoyu Chen, Tingting Wang

Published Aug 1, 2015• 2 citations
Publisher

Abstract

An N -qubit Greenberger–Horne–Zeilinger (GHZ) state has many applications in various quantum information tasks and can be realized in different experimental schemes. A GHZ diagonal state evolves to another GHZ diagonal state in independent parallel Pauli channels. We give the explicit expression of the resultant GHZ diagonal state in terms of the initial state and channel parameters. If the initial state is a pure N qubit GHZ state or a three-qubit GHZ diagonal state admits a condition, the full separability criterion of the Pauli noisy state is equivalent to positive partial transpose (PPT) criterion. Thus the fully separable condition follows.

DOI
10.1088/1674-1056/24/8/080303

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An N -qubit Greenberger–Horne–Zeilinger (GHZ) state has many applications in various quantum information tasks and can be realized in different experimental schemes. A GHZ diagonal state evolves to another GHZ diagonal state in independent parallel Pauli channels. We give the explicit expression of the resultant GHZ diagonal state in terms of the initial state and channel parameters. If the initial state is a pure N qubit GHZ state or a three-qubit GHZ diagonal state admits a condition, the full separability criterion of the Pauli noisy state is equivalent to positive partial transpose (PPT) criterion. Thus the fully separable condition follows.
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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 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.
We analytically prove the necessary and sufficient criterion for the full separability of three-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states. The corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others using the relative entropy of entanglement. We show that the biseparable criterion and the genuine entanglement are determined only by the biggest GHZ diagonal element regardless of all the other smaller diagonal elements. We have completely solved the entanglement problems of three-qubit GHZ diagonal states.
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We give separability criteria for general multi-qubit states in terms of diagonal and anti-diagonal entries. We define two numbers which are obtained from diagonal and anti-diagonal entries, respectively, and compare them to get criteria. They give rise to characterizations of separability when all the entries are zero except for diagonal and anti-diagonal, like Greenberger-Horne-Zeilinger diagonal states. The criteria is strong enough to get nonzero volume of entanglement with positive partial transposes.