The negativityquant-ph/0102117 is an entanglement measure which is easy to compute.

The negativity can be defined as:

$$\mathcal{N}(\rho) := \frac{||\rho^{\Gamma_A}||_1-1}{2}$$

where:

• ρ^Γ_A is the partial transpose of ρ with respect to subsystem A
• $||X||_1 = Tr|X| = Tr \sqrt{X^\dagger X}$ is the trace norm or the sum of the sigular values of the operator X.

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of ρ^Γ_A:

$$\mathcal{N}(\rho) := \sum_i \frac{|\lambda_{i}|-\lambda_{i}}{2}$$
where λ_i are all of the eigenvalues.

Properties

• Is a convex function of ρ:

N(∑_ip_iρ_i) ≤ ∑_ip_iN(ρ_i)

• Is an entanglement monotone:

N(P(ρ_i)) ≤ N(ρ_i)

where P(ρ) is an arbitrary LOCC operation over ρ

See also

• Logarithmic negativity

Category:Quantum Information Theory Category:Handbook of Quantum Information Category:Entanglement