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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

where $Γ A$ is the partial transpose operation and $|| \cdot ||_1$ denotes the trace norm.

It relates to the negativity as follows:

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: $E_N(\rho \otimes \sigma) = E_N(\rho) \cdot E_N(\sigma)$
is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces $H_1, H_2, \ldots$ (typically with increasing dimension) we can have a sequence of quantum states $\rho_1, \rho_2, \ldots$ which converges to $\rho^{\otimes n_1}, \rho^{\otimes n_2}, \ldots$ (typically with increasing $n i$ ) in the trace distance, but the sequence $E_N(\rho_1)/n_1, E_N(\rho_2)/n_2, \ldots$ does not converge to $E N (ρ)$ .
is an upper bound to the distillable entanglement