Construction of genuine multipartite entangled states

Abstract

Genuine multipartite entanglement is of great importance in quantum information, especially from the experimental point of view. Nevertheless, it is difficult to construct genuine multipartite entangled states systematically, because the genuine multipartite entanglement is unruly. We propose another product based on the Kronecker product in this paper. The Kronecker product is a common product in quantum information with good physical interpretation. We mainly investigate whether the proposed product of two genuine multipartite entangled states is still a genuine entangled one. We understand the entanglement of the proposed product better by characterizing the entanglement of the Kronecker product. Then we show the proposed product is a genuine multipartite entangled state in two cases. The results provide a systematical method to construct genuine multipartite entangled states of more parties. Contents

Inertias of entanglement witnesses

“…we can take ρ (AC):(BD) as a tripartite state of system (AC), B, D, i.e., ρ (AC):B:D , and take ρ Γ (AC):(BD) as the partial transpose of the tripartite state ρ (AC):B:D with respect to subsystem (AC), i.e., ρ Γ AC (AC):B:D . In this way we can construct the tripartite genuinely entangled state ρ (AC):B:D using two bipartite entangled states α AB and β CD [17]. Moreover, we conjectured in [17] that ρ (AC):B:D is a tripartite genuinely entangled state if both α AB and β CD are entangled.…”

“…Moreover, we conjectured in [17] that ρ (AC):B:D is a tripartite genuinely entangled state if both α AB and β CD are entangled. We have shown the above conjecture holds if either R(α AB ) or R(β CD ) is not spanned by product vectors [17]. The latest progress on this conjecture has been made in [59].…”

“…As we know, genuine mulitpartite entanglement is valuable resouce in quantum information processing tasks [6,[60][61][62]. Nevertheless, it is difficult to characterize genuinely multipartite entangled (GME) states [16,17]. Obviously, the characterization of the partial transpose of GME state is also hard.…”

“…It has been shown that determining whether a bipartite state is entangled or not is an NP-hard problem [15]. It is even harder to tame multipartite entanglement [16], since multipartite entangled states can be further classified as genuinely multipartite entangled states and biseparable states [17]. In 2000, Terhal first introduced the term entanglement witness (EW) by indicating that a violation of a Bell inequality can be expressed as a witness for entanglement [18].…”

Detection of genuine tripartite entanglement by two bipartite entangled states

“…One may show that Conjecture 3 (ii) is a special case of (i). We first consider to attack the generic one in Theorem 4, by introducing a known result from [42]. (i) When p 1 = p 2 = −1, the region of the GE detection of state ρ is maximum for 0.902646 < x ≤ 1.…”