Bound entanglement and distillability of multipartite quantum systems

Abstract

We construct a class of entangled states in $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ quantum systems with $dim\mathcal{H}_{A}=dim\mathcal{H}_{B}=dim\mathcal{H}_{C}=2$ and classify those states with respect to their distillability properties. The states are bound entanglement for the bipartite split$(AB)-C$. The states are NPT entanglement and $1$-copy undistillable for the bipartite splits $A-(BC)$ and $B-(AC)$. Moreover, we generalize the results of $2\otimes2\otimes2$ systems to the case of $2n\otimes 2n\otimes2n$ systems

Non-Markovian dynamics of quantum open systems embedded in a hybrid environment

“…Actually, this special case is very common in many interesting models [26,33,27,15] in which the exact O or Q operators just contain no noises. Moreover, in general case, we can still expand O and Q into functional series and only taking the first term (with zeroth order of noise variables) of the expansions as…”

“…Fundamentally, the dynamics of open quantum systems embedded in one or more environments has attracted the wide-spread interest in recent years [5,6,7,8]. On the one hand, the temporal behaviours of quantum open systems are essential for understanding many fundamental issues of quantum theory such as quantum dissipation and decoherence [9,10,11,12,13,14,15]. On the other hand, many novel applications based on quantum devices also require a better understanding on the interaction between the quantum system of interest and its environment in order to manipulate and control the system's dynamics [16,17].…”

“…For the case of bosonic bath, several interesting physical systems have been studied in the past fifteen years [26,27,28,36,37]. As a fundamentally theoretical study, it has been developed from solving single-particle system to solving many-body systems [8,14,15]. Furthermore, as a computing tool in real applications [38,39], the NMQSD approach also showed its potential value in many interesting problems including precision quantum measurement [40], quantum control dynamics [41], and quantum biology [42].…”

Construction of bound entangled states based on permutation operators

“…In fact, since ρ is PPT, ρ ′ is also PPT. By [18], the following vectors form a basis of the range of ρ: |ψ 3m−2,3m−2 = (0, · · · , 0, a 3m−2 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3m−2 , 0, · · · , 0) t , |ψ 3m−2,3n−2 = (0, · · · , 0, a 3m−2 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n−2 , 0, · · · , 0) t , |ψ 3m−2,3n−1 = (0, · · · , 0, a 3m−2 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n−1 , 0, · · · , 0) t , |ψ 3m−2,3n = (0, · · · , 0, a 3m−2 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n , 0, · · · , 0) t , |ψ 3m−1,3m−1 = (0, · · · , 0, a 3m−1 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3m−1 , 0, · · · , 0) t , |ψ 3m−1,3n−2 = (0, · · · , 0, a 3m−1 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n−2 , 0, · · · , 0) t , |ψ 3m−1,3n−1 = (0, · · · , 0, a 3m−1 , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n−1 , 0, · · · , 0) t , |ψ 3m,3m = (0, · · · , 0, a 3m , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3m , 0, · · · , 0) t , |ψ 3m,3n−2 = (0, · · · , 0, a 3m , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n−2 , 0, · · · , 0) t , |ψ 3m,3n = (0, · · · , 0, a 3m , 0, · · · , 0) t ⊗ (0, · · · , 0, b 3n , 0, · · · , 0) t ,…”

“…In fact, since ρ is PPT, ρ ′ is also PPT. By [18], the following vectors form a basis of the range of ρ:…”