# Negativity

The negativity1 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:

• ρΓA 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 ρΓA:

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

Properties

N(∑ipiρi) ≤ ∑ipiN(ρi)

N(P(ρi)) ≤ N(ρi)

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