# Logarithmic negativity

The **logarithmic negativity**1 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 2 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
*H*1,*H*2, … (typically with increasing dimension) we can have a sequence of quantum states*ρ*1,*ρ*2, … which converges to*ρ*⊗*n*1,*ρ*⊗*n*2, … (typically with increasing*n**i*) in the trace distance, but the sequence*E**N*(*ρ*1)/*n*1,*E**N*(*ρ*2)/*n*2, … does not converge to*E**N*(*ρ*). - is an upper bound to the distillable entanglement

### See also

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

Last modified:

Monday, October 26, 2015 - 17:56