Separability of three qubit Greenberger–Horne–Zeilinger diagonal states

Abstract

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 [1] 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. IntroductionEntanglement is now considered as one of the most important resources in quantum information theory, and it is crucial to detect entanglement from separability. Positivity of partial transposes is a simple but powerful criterion for separability [2,3]. In fact, it is known [4,5] that PPT property is equivalent to separability for 2⊗2 or 2⊗3 systems. But, it is very difficult in general to distinguish separability from entanglement, as it is known to be an NP -hard problem [6].The purpose of this note is to give a complete characterization of separability for three qubit Greenberger-Horne-Zeilinger diagonal states, which are diagonal in the GHZ basis. Those include mixtures of GHZ states and identity, as they were considered in [7,8] for examples. In multi-qubit systems, GHZ states [9,10] are key examples of maximally entangled states, and they are known to have many applications in quantum information theory. They also play central roles in the classification of multi-qubit entanglement [7,11,12]. See survey articles [13,14] for general theory of entanglement as well as various aspects of GHZ states.Our main tool is to use the notion of entanglement witnesses. In the bi-partite cases, positive linear maps are very useful to detect entanglement through the duality between tensor products and linear maps [5,15]. This duality has been formulated as the notion of entanglement witnesses [16], which is still valid in multi-partite cases. In the tri-partite cases, the second author [17] has interpreted entanglement witnesses as positive bi-linear maps. With this interpretation, we carefully choose useful entanglement witnesses for our purpose.1991 Mathematics Subject Classification. 81P15, 15A30, 46L05, 46L07.

Absolutely separating quantum maps and channels

“…. The criterion of full separability is known, for instance, for 3-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states [34,35].…”

“…where the binary representation of -= (9) is GHZ diagonal, so we apply to it the necessary and sufficient condition of full separability( [35], theorem 5.2), which shows that (9) is not fully separable. Thus,…”

Separability criterion for three-qubit states with a four dimensional norm

“…For example, separability for 2 ⊗ 2, 2 ⊗ 3 states and 2 ⊗ n states with low ranks is known to be equivalent to positivity of partial transposes [1][2][3][4][5][6]. In the three qubit cases, separability of Greenberger-Horne-Zeilinger diagonal states has been completely characterized recently by the second and third authors [7]. Note that separability problem is known to be an NP-hard problem in general [8].…”

“…The purpose of this paper is to provide a complete characterization for the separability of three qubit X-states. Because the X-part of a separable three qubit state is again separable [7], our characterization gives rise to a necessary separability criterion in terms of diagonal and anti-diagonal entries for general three qubit states.…”

“…On the other hand, the second and third authors gave a characterization of three qubit X-shaped entanglement witnesses [17]. They also proved [7] that a complicated expression in the Gühne's criterion can be replaced by a simpler formula in [17], and the Gühne's criterion is actually sufficient for separability of GHZ diagonal states.…”