We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of $n-1$ particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of $n-1$ particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.

Jun 15, 200089 citations

On the necessary and sufficient conditions for separability of quantum mixed states

Michał Horodecki, Ryszard Horodecki, Paweł Horodecki

A state acting on Hilbert space {\\cal H}_1\\otimes{\\cal H}_2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2\\times2 and 2\\times3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability.A state acting on Hilbert space {\\cal H}_1\\otimes{\\cal H}_2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2\\times2 and 2\\times3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability.

May 28, 19966 citations

Separability Criteria for Quantum Mixed States in Terms of Trace Norm

赵慧

Chinese Physics LettersJan 1, 20060 citations

Criteria for non-k-separability of n-partite quantum states

N. Ananth, V. K. Chandrasekar, Murugaian Senthilvelan

arXiv:1501.04162The European Physical Journal DFeb 1, 201515 citations

Separability of mixed quantum states: linear contractions approach

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

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.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.

arXiv:quant-ph/0206008Jun 2, 200217 citations

We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of

n-1

particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.