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Journal of Physics A Mathematical and Theoretical

LPP elementary operator criterion of full separability for states in multipartite quantum systems

Siqing Yan, Jinchuan Hou

Published Oct 10, 2012• 5 citations
Publisher

Abstract

Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.

DOI
10.1088/1751-8113/45/43/435303

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Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.
Let H and K be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from into is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is given which shows that a state ρ on H ⊗ K is separable if and only if (Φ⊗ I )ρ ⩾ 0 for all positive finite-rank elementary operators Φ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.
We show that multi-invariants provide necessary and sufficient criteria for full separability in finite-dimensional multipartite systems. For tripartite three-qubit states, the third-order negativity gives an exact criterion for full separability for both pure and mixed states. We then extend the framework to four-qubit pure states, whose full characterization requires eighteen independent real parameters. We identify eighteen independent entanglement measures, consisting of six bipartite, eight tripartite, and four quadripartite measures, with the third-order negativity serving as a key separability criterion. Finally, we generalize the construction to multipartite qudit systems and discuss an application to conformal field theory.
Let HHH and KKK be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from B(H){\mathcal B}(H)B(H) into B(K){\mathcal B}(K)B(K) is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state ρ\rhoρ on H⊗KH\otimes KH⊗K is separable if and only if (Φ⊗I)ρ≥0(\Phi\otimes I)\rho\geq 0(Φ⊗I)ρ≥0 for all positive finite rank elementary operators Φ\PhiΦ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.
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.
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.