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Journal of Beijing University of Technology

Entanglement Properties of the Tripartite Quantum States

Zhao Hu

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

To study the entanglement properties of multipartite quantum states,a class of states in the tripartite quantum systems was constructed.Using the properties of the partitioned matrix,the positive matrix and the diagonally dominant matrix,applying the range criterion,the separable criterion,the structure and the partial transpositions of their density matrices were analyzed.The research shows that the states are entangled for the splits A-BC and B-AC,separable for the split AB-C.Therefore it can be concluded that the states are bound entangled for the splits A-BC and B-AC,separable with respect to the split AB-C,and finally the entanglement properties of any bipartite splits of the tripartite states are provided.

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To study the entanglement properties of multipartite quantum states,a class of states in the tripartite quantum systems was constructed.Using the properties of the partitioned matrix,the positive matrix and the diagonally dominant matrix,applying the range criterion,the separable criterion,the structure and the partial transpositions of their density matrices were analyzed.The research shows that the states are entangled for the splits A-BC and B-AC,separable for the split AB-C.Therefore it can be concluded that the states are bound entangled for the splits A-BC and B-AC,separable with respect to the split AB-C,and finally the entanglement properties of any bipartite splits of the tripartite states are provided.
We construct a class of entangled states in ℋ = ℋA ⊗ ℋB ⊗ ℋC quantum systems with dim ℋA = dim ℋB = dim ℋC = 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 non-positive partial transpose (NPT) entanglement and 1-copy undistillable for the bipartite splits A - (BC) and B - (AC). Moreover, we generalize the results of 2 ⊗ 2 ⊗ 2 systems to the case of 2n ⊗ 2n ⊗ 2n systems.
In this paper, we investigate the genuine entanglement in tripartite systems based on partial transposition and the norm of correlation tensors of the density matrices. We first derive an analytical sufficient criterion to detect genuine entanglement of tripartite qubit states combining with the partial transposition of the density matrices. Then, we use the norm of correlation tensors to study genuine entanglement for tripartite qudit quantum states and obtain a genuine entanglement criterion by constructing certain matrices. With detailed examples, our results are seen to be able to detect more genuine tripartite entangled states than previous studies.
State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say, Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC)] is a strict subset of the set of states having positive partial transposition across the three bipartite cuts [say, Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC)] for all the tripartite Hilbert spaces ${\mathbb{C}}_{A}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}_{B}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}_{C}^{{d}_{3}}$ with $min{{d}_{1},{d}_{2},{d}_{3}}\ensuremath{\ge}2$. The claim is proved by constructing a state belonging to the set Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC) but not to Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC). For ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}3}$ with $d\ensuremath{\ge}3$, the construction follows from a specific type of multipartite unextendible product bases. However, such a construction is not possible for ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}3}$ since bipartite systems ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ do not allow any unextendible product bases for arbitrary nnn [Phys. Rev. Lett. 82, 5385 (1999)]. For the three-qubit system, we therefore come up with a different construction.
Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.