# Logarithmic negativity

The logarithmic negativityquant-ph/0505071 is an entanglement measure which is easily computable and an upper bound to the distillable entanglement. It is defined as

EN(ρ) :  = log2∣∣ρΓA∣∣1
where ΓA is the partial transpose operation and ∣∣ ⋅ ∣∣1 denotes the trace norm.

It relates to the negativity quant-ph/0102117 as follows:

EN(ρ) :  = log2(2N + 1).

### Properties

The logarithmic negativity

• can be zero even if the state is entangled (if the state is PPT entangled)
• does not reduce to the entropy of entanglement on pure states like most other entanglement measures
• is additive on tensor products: EN(ρ ⊗ σ) = EN(ρ) ⋅ EN(σ)
• is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces H1, H2, … (typically with increasing dimension) we can have a sequence of quantum states ρ1, ρ2, … which converges to ρ ⊗ n1, ρ ⊗ n2, … (typically with increasing ni) in the trace distance, but the sequence EN(ρ1)/n1, EN(ρ2)/n2, … does not converge to EN(ρ).
• is an upper bound to the distillable entanglement