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arXiv:quant-ph/9912010 · Physical Review A

Separability in 2×N composite quantum systems

Barbara Kraus, J. I. Cirac, Siniša Karnas, Maciej Lewenstein

Published May 8, 2000• 159 citations
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Abstract

We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed $3N.$ If it exceeds this number, we show that one can subtract product vectors until it decreases to $3N,$ while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.

DOI
10.1103/physreva.61.062302

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We analyze the separability properties of density operators supported on C2\ensuremath⨂CN{C}^{2}\ensuremath{\bigotimes}{C}^{N}C2\ensuremath⨂CN whose partial transposes are positive operators. We show that if the rank of \ensuremathρ\ensuremath{\rho}\ensuremathρ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N.3N.3N. If it exceeds this number, we show that one can subtract product vectors until it decreases to 3N,3N,3N, while keeping the positivity of \ensuremathρ\ensuremath{\rho}\ensuremathρ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.
We show that all density operators of 2×N\times N×N--dimensional quantum systems that remain invariant after partial transposition with respect to the first system are separable. Based on this criterion, we derive a sufficient separability condition for general density operators in such quantum systems. We also give a simple proof of the separability criterion in 2×22\times 22×2--dimensional systems [A. Peres, Phys. Rev. Lett {\bf 77}, 1413 (1996)]
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We investigate separability and entanglement of mixed states in ${C}^{2}\ensuremath{\bigotimes}{C}^{2}\ensuremath{\bigotimes}{C}^{N}$ three party quantum systems. We show that all states with positive partial transposes that have rank< N\mathrm{rank}<~Nrank< N are separable. For the three-qubit case (N=2)(N=2)(N=2) we prove that all states \ensuremath{\rho} that have positive partial transposes and rank three are separable. We also provide constructive separability checks for the states \ensuremath{\rho} that have the sum of the rank of \ensuremath{\rho} and the ranks of partial transposes with respect to all subsystems smaller than $15N\ensuremath{-}1.$
Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for noncommuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit moment matrix (based on PPT constraints on bipartite divisions of the density matrix) and show that three-qubit pure states with nonzero tangle violate these PPT moment constraints. Further, we recover a necessary and sufficient condition of separability in a multiqubit Werner state through PPT bounds on moments.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
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.