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arXiv:quant-ph/9911044 · Physical Review A

Classification of multiqubit mixed states: Separability and distillability properties

Wolfgang Dür, J. I. Cirac

Published Mar 17, 2000• 291 citations
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Abstract

We give a complete, hierarchic classification for arbitrary multiqubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multiqubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.

DOI
10.1103/physreva.61.042314

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We give a complete, hierarchic classification for arbitrary multiqubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multiqubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.
We present a family of 3-qubit states to which any arbitrary state can be depolarized. We fully classify those states with respect to their separability and distillability properties. This provides a sufficient condition for nonseparability and distillability for arbitrary states. We generalize our results to NNN-particle states.
We extend the classification of mixed states of quantum systems composed of arbitrary numbers of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+11+18+11+18+1 partial separability classes in the tripartite case contrary to a former 1+8+11+8+11+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.
We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given mmm-partite subsystem with mmm qubits and the purity of states of this mmm-partite subsystem for a pure state in multipartite systems with arbitrary finite n(>m)n(>m)n(>m) qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite systems.
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.
We analyze 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 analyze a criterion of separability that 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 that distills any inseparable two-qubit state. In particular, this means that any state $\ensuremath{\varrho}$ of two N-level systems with $〈{\ensuremath{\psi}}_{+}|\ensuremath{\varrho}|{\ensuremath{\psi}}_{+}〉>1/N$ can be distilled $({\ensuremath{\psi}}_{+}$ is the singlet state generalized to higher dimension). The criterion also singles out all the states that can be distilled by a class of protocols. The proof involves construction of the family of states that are invariant under transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varrho}}U\ensuremath{\bigotimes}{U}^{*}\ensuremath{\varrho}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}{U}^{*\ifmmode\dagger\else\textdagger\fi{}},$ where U is a unitary transformation and the asterisk denotes complex conjugation. The states are related to the depolarizing channel generalized to the nonbinary case.