In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability. It has been shown to be an entanglement monotone and hence a proper measure of entanglement.
Definition The negativity of a subsystem A {\displaystyle A} can be defined in terms of a density matrix ρ {\displaystyle \rho } as:
N ( ρ ) ≡ | | ρ Γ A | | 1 − 1 2 {\displaystyle {\mathcal {N}}(\rho )\equiv {\frac {||\rho ^{\Gamma _{A}}||_{1}-1}{2}}}
where:
ρ Γ A {\displaystyle \rho ^{\Gamma _{A}}} is the partial transpose of ρ {\displaystyle \rho } with respect to subsystem A {\displaystyle A}
| | X | | 1 = Tr | X | = Tr X † X {\displaystyle ||X||_{1}={\text{Tr}}|X|={\text{Tr}}{\sqrt {X^{\dagger }X}}} is the trace norm or the sum of the singular values of the operator X {\displaystyle X} . An alternative and equivalent definition is the absolute sum of the negative eigenvalues of ρ Γ A {\displaystyle \rho ^{\Gamma _{A}}} :
N ( ρ ) = | ∑ λ i < 0 λ i | = ∑ i | λ i | − λ i 2 {\displaystyle {\mathcal {N}}(\rho )=\left|\sum _{\lambda _{i}<0}\lambda _{i}\right|=\sum _{i}{\frac {|\lambda _{i}|-\lambda _{i}}{2}}}
where λ i {\displaystyle \lambda _{i}} are all of the eigenvalues.
Properties Is a convex function of ρ {\displaystyle \rho } :
N ( ∑ i p i ρ i ) ≤ ∑ i p i N ( ρ i ) {\displaystyle {\mathcal {N}}(\sum _{i}p_{i}\rho _{i})\leq \sum _{i}p_{i}{\mathcal {N}}(\rho _{i})}
Is an entanglement monotone:
N ( P ( ρ ) ) ≤ N ( ρ ) {\displaystyle {\mathcal {N}}(P(\rho ))\leq {\mathcal {N}}(\rho )}
where P ( ρ ) {\displaystyle P(\rho )} is an arbitrary LOCC operation over ρ {\displaystyle \rho }
Logarithmic negativity The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement. It is defined as
E N ( ρ ) ≡ log 2 | | ρ Γ A | | 1 {\displaystyle E_{N}(\rho )\equiv \log _{2}||\rho ^{\Gamma _{A}}||_{1}}
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