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The reduction criterionquant-ph/9708015 is a
separability
criterion, that is a condition that all separable states have to
satisfy and a violation of it is therefore a proof of entanglement.
Let ρA := trB(ρAB)
and ρB := trA(ρAB).
Then the reduction criterion states that for any separable ρAB,
two derived operators must be positive semidefinite:
ρA ⊗ IB − ρAB ≥ 0
IA ⊗ ρB − ρAB ≥ 0.
The criterion comes from applying the positive but not completely
positive (PnCP) map
L(ρ) = I tr(ρ) − ρ
to one part of a
bipartite
system.
In general it is weaker than the
PPT
criterion, but any state violating it is always
distillable.
For states of dimension 2 × 2 or 2 × 3 it is equivalent to the PPT criterion
and therefore also necessary and sufficient.
All entangled
Werner
states of local dimension above 3 satisfy the reduction criterion,
but violate the stronger PPT criterion. All entangled
isotropic
states violate the reduction criterion.
\bibitem{quant-ph/9708015}
Category:Entanglement
Category:Handbook
of Quantum Information
Last modified:
Monday, October 26, 2015 - 17:56
