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

E_N(ρ) := log₂||ρ^Γ_A||₁ where Γ_A is the partial transpose operation and || ⋅ ||₁ denotes the trace norm.

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

E_N(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: E_N(ρ ⊗ σ) = E_N(ρ) ⋅ E_N(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 n_i) in the trace distance, but the sequence E_N(ρ₁)/n₁, E_N(ρ₂)/n₂, … does not converge to E_N(ρ).
• is an upper bound to the distillable entanglement

See also

• Negativity

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