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

Communications in Theoretical Physics

Separability of Rank-Two Quantum States in $C^M \otimes C^N$ Composite Quantum Systems

Shao-Ming Fei, Xiu-Hong Gao, Xiaohong Wang, Zhi‐Xi Wang, Ke Wu

Published May 15, 2003• 2 citations
Publisher

Abstract

We consider rank-two density matrices ρ supported on an M × N Hilbert space for arbitrary dimensions M and N . Explicit sufficient and necessary conditions for separability of ρ are presented.
DOI
10.1088/0253-6102/39/5/525

Related papers

Separability of Rank—Two Quantum States in C^M ×CN Composite Quantum Systems

FEIShao-Ming, GAOXiu-Hong, WANGXiao-Hong +2 more

We consider rank-two density matrices ρ supported on an M × N Hilbert space for arbitrary dimensions M and N. Explicit sufficient and necessary conditions for separability of ρ are presented.

理论物理通讯:英文版•Jan 1, 2003•0 citations

Canonical Form and Separability of PPT States in 2xMxN Composite Quantum Systems

X. H. Wang, S. M. Fei, Z. X. Wang +1 more

We investigate the canonical forms of positive partial transposition (PPT) density matrices in ${\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^N$ composite quantum systems with rank $N$. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.

arXiv:quant-ph/0312031•Dec 3, 2003•0 citations

SEPARABILITY OF RANK TWO QUANTUM STATES ON MULTIPLE QUANTUM SPACES WITH DIFFERENT DIMENSIONS

Shao-Ming Fei, Xiu-Hong Gao, Xiaohong Wang +2 more

We consider the separability of rank two quantum states on multiple quantum spaces with different dimensions. The sufficient and necessary conditions for separability of these multiparty quantum states are explicitly presented. A nonseparability inequality is also given for the case where one of the eigenvectors corresponding to nonzero eigenvalues of the density matrix is maximally entangled.

arXiv:quant-ph/0305080•International Journal of Quantum Information•Mar 1, 2003•5 citations

CANONICAL FORM AND SEPARABILITY OF PPT STATES IN ${\mathcal C}^2\otimes {\mathcal C}^M\otimes {\mathcal C}^N$ COMPOSITE QUANTUM SYSTEMS

Xiaohong Wang, Shao-Ming Fei, Zhi‐Xi Wang +1 more

We investigate the canonical forms of positive partial transposition (PPT) density matrices in [Formula: see text] composite quantum systems with rank N. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.

International Journal of Quantum Information•Sep 1, 2003•0 citations

Multiparticle entanglement and ranks of density matrices

Joachim Stolze, Bo Chong, Hellmut Keiter

Based on the ranks of reduced density matrices, we derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement. In a similar way we obtain necessary conditions for the separability of a given mixed state with respect to partitions of all particles of the system into subsets. The special case of pure states is discussed separately.

arXiv:quant-ph/0512199•Jan 1, 2005•1 citation

Operational criterion and constructive checks for the separability of low-rank density matrices

Paweł Horodecki, Maciej Lewenstein, Guifré Vidal +1 more

We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N $(M<~N),$ and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}>~0.$ For rank $r(\ensuremath{\varrho})<~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})<~2MN\ensuremath{-}M\ensuremath{-}N+2.$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.

arXiv:quant-ph/0002089•Physical Review A•Aug 17, 2000•150 citations
Separability of Rank-Two Quantum States in CM⊗CNC^M \otimes C^NCM⊗CN Composite Quantum Systems
We investigate the canonical forms of positive partial transposition (PPT) density matrices in C2⊗CM⊗CN{\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^NC2⊗CM⊗CN composite quantum systems with rank NNN. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.
CANONICAL FORM AND SEPARABILITY OF PPT STATES IN C2⊗CM⊗CN{\mathcal C}^2\otimes {\mathcal C}^M\otimes {\mathcal C}^NC2⊗CM⊗CN COMPOSITE QUANTUM SYSTEMS
We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N (M< N),(M<~N),(M< N), and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}>~0.$ For rank $r(\ensuremath{\varrho})<~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})<~2MN\ensuremath{-}M\ensuremath{-}N+2.$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.