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Physical Review A

Reduction criterion for separability

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

Published Aug 1, 1999• 173 citations

Abstract

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.

DOI: 10.1103/physreva.60.898

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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.

A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.

A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights

{w}_{A}

satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.

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

\varrho\to U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}

where

U

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.