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A separability criterion for density operators

Rudolph, O

Published Jan 1, 2000• 91 citations
OpenAlex

Abstract

We give a necessary and sufficient condition for a mixed quantum mechanicalstate to be separable. The criterion is formulated as a boundedness conditionin terms of the greatest cross norm on the tensor product of trace classoperators.

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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.
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.