Two-way and three-way negativities of three-qubit entangled states

Abstract

In this letter we propose to quantify three qubit entanglement using global negativity along with K−way negativities, where K = 2 and 3. The principle underlying the definition of K−way negativity for pure and mixed states of N −subsystems is PPT sufficient condition. However, K−way partial transpose with respect to a subsystem is defined so as to shift the focus to K−way coherences instead of K subsystems of the composite system. PACS numbers:Quantum entanglement is not only a fascinating aspect of multipartite quantum systems, but also a physical resource needed for quantum communication, quantum computation and information processing in general. Bipartite entanglement is well understood, however, many aspects of multipartite entanglement are still to be investigated. Peres [1] and the Horedecki [2,3,4] have shown a positive partial transpose (PPT) of a bipartite density operator to be a sufficient criterion for classifying bipartite entanglement. Negativity [5,6] based on Peres Horodecski criterion has been shown to be an entanglement monotone [7,8,9]. Negativity is a useful concept being related to the eigenvalues of partially transposed state operator and can be calculated easily. In this letter, we define 2−way and 3−way negativities and propose a classification of three qubit states based on measures related to global, 2−way and 3−way negativities. General definition of K−way negativities for pure and mixed states of N −subsystems is given in ref [10]. The K−way partial transpose with respect to a subsystem is defined so as to shift the focus to K−way coherences instead of K subsystems of the composite system. While pure K− partite entanglement of a composite system is generated by K−way coherences, K−partite entanglement can, in general, be present due to (K − 1) way coherences as well. I. GLOBAL NEGATIVITYThe Hilbert space, To measure the overall entanglement of a subsystem p, we shall use twice the negativity as defined by Vidal and Werner [6], and call it global Negativity N (2) * Electronic address: shelly@uel.br † Electronic address: nsharma@uel.br

PartialK-way negativities and three-tangle for three-qubit states

“…An interesting pure state for which tripartite entanglement has been studied by Lohmayer et al [12] is a superposition of three qubit GHZ state and W state. In our paper [13], we calculated N A G , E A 3 , and E A 2 for single parameter pure states…”

“…To put the record straight, the states Ψ (±) are not in canonical form. Therefore the difference ∆ = E A 3 N A G − τ 3 for a given state, observed in [13] measures the distance of the state from the canonical state in terms of excess or deficit of three qubit coherences in comparison with that for the canonical state. The states This analysis leads to a simple explanation for why a GHZ state can not be converted to a W-state.…”

Sequential generation of polynomial invariants and N-body non-local correlations

“…Appendix B is a note on transvection process to obtain invariants of a binary form. The principal construction tools that is local unitary (LU) invariance, selective partial transposition [31,32] and notion of negativity fonts, used in our earlier works [18,19], are also needed for sequential construction of polynomial invariants that quantify N -way correlations. For the sake of completeness, we define the determinants of negativity fonts in section IV.…”

Entanglement dynamics and Bell violation of three-qubit states coupled to an XY spin chain with Dzyaloshinski–Moriya interaction

“…Negativity can be used to access the quantum entanglement of both pure and mixed states efficiently. Due to its operational and calculational properties, negativity has been employed to evaluate quantum entanglement of multipartite quantum states with high dimensions extensively [7][8][9][10].…”