Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics B

A sufficient and necessary criterion for separability of bipartite quantum states

Hui Zhao, Xinyu Yu

Published Jul 31, 2014• 0 citations
Publisher

Abstract

A sufficient and necessary criterion for separability of bipartite quantum states is presented. We construct a class of mixed states in 2 ⊗ k quantum systems and in 3 ⊗ k quantum systems, respectively and show that these states are separable if and only if they are positive partial transpositions.
DOI
10.1088/1674-1056/23/8/080302

Related papers

A Separability of mixed states: necessary and sufficient conditions

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

We provide necessary and sufficient conditions for the separability of mixed states. As a result we obtain a simple criterion of the separability for 2 x 2 and 2 x 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, this is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.

•Jan 1, 1996•2,276 citations

On the necessary and sufficient conditions for separability of quantum mixed states

Michał Horodecki, Ryszard Horodecki, Paweł Horodecki

A state acting on Hilbert space {\\cal H}_1\\otimes{\\cal H}_2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2\\times2 and 2\\times3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability.

•May 28, 1996•6 citations

A class of separable quantum states

Yu Jing Guo, Jinchuan Hou

We present a sufficient condition for a bipartite state to be separable via the strong positive partial transposition structure introduced in Chru?ci?ski et?al (2008 Phys. Rev. A 77 022113), from which we obtain a class of separable states. In particular, we prove that any classical-quantum state is a state of our class but not vice versa. Both finite- and infinite-dimensional systems are considered.

Journal of Physics A Mathematical and Theoretical•Nov 21, 2012•16 citations

A NOTE ON QUANTUM ENTANGLEMENT AND PPT

Shao-Ming Fei, Xianqing Li‐Jost

We study quantum states for which the PPT criterion is both sufficient and necessary for separability. We present a class of 3 × 3 bipartite mixed states and show that these states are separable if and only if they are PPT.

arXiv:0904.4766•International Journal of Quantum Information•Apr 1, 2009•5 citations

Separability of mixed quantum states: linear contractions approach

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely triparite entanglement. We also show that for multipartite system any permutation of indices of density matrix leads to separability criterion.

arXiv:quant-ph/0206008•Jun 2, 2002•17 citations

Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.

Open Systems & Information Dynamics•Mar 1, 2006•139 citations
A state acting on Hilbert space {\\cal H}_1\\otimes{\\cal H}_2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2\\times2 and 2\\times3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability.
Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely triparite entanglement. We also show that for multipartite system any permutation of indices of density matrix leads to separability criterion.
Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.