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Reduction criterion for separability

1999; American Physical Society; Volume: 60; Issue: 2 Linguagem: Inglês

10.1103/physreva.60.898

1538-4446

Nicolas J. Cerf, Christoph Adami, Robert M. Gingrich,

Quantum Mechanics and Applications

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.