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arXiv:quant-ph/0312185 · Physical Review A

Generalized reduction criterion for separability of quantum states

Sergio A Albeverio, Kai Chen, Shao-Ming Fei

Published Dec 17, 2003• 63 citations
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Abstract

A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.

DOI
10.1103/physreva.68.062313

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A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.
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.
The recent proposed realignment separability criterion for mixed is analyzed. We identify the essential part of this criterion is a swap operator followed by a partial transposition. Then we analyze the separability criterion of permutation indices of the density matrix for multipartite state which is a generalization of the realignment criterion. We give a method to solve the equivalence problem of the separability criterion of permutation indices of the density matrix. For tripartite state, we show the non-trivial separability criteria are either partial transposition criterion or realignment criterion.
We introduce a separability criterion based on the positive map $\ensuremath{\Gamma}:\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}(\mathrm{Tr}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho},$ where $\ensuremath{\rho}$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\ensuremath{\Gamma}$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\ensuremath{\Gamma}$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\ifmmode\times\else\texttimes\fi{}2$ and $2\ifmmode\times\else\texttimes\fi{}3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the ``magic'' basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.
We analyse the problem of distillation of entanglement of mixed states in higher dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223 1 (1996)] we introduce and analyse a criterion of separability which relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol which distills any inseparable two-qubit state. Conversely, all the states which can be distilled by such a protocol must violate the criterion. The proof involves construction of the family of states which are invariant under transformation ϱ→U⊗U∗ϱU†⊗U∗†\varrho\to U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}ϱ→U⊗U∗ϱU†⊗U∗† where UUU is a unitary transformation and star denotes complex conjugation. The states are related to the depolarizing channel generalized to non-binary case.