Logarithmic negativity

The '''logarithmic negativity'''quant-ph/0505071 is an [[entanglement measure]] which is easily computable and an upper bound to the [[distillable entanglement]]. It is defined as :$E\_N(\backslash rho)\; :=\; \backslash log\_2\; ||\backslash rho^\{\backslash Gamma\_A\}||\_1$ where $\backslash Gamma\_A$ is the [[partial transpose]] operation and $||\; \backslash cdot\; ||\_1$ denotes the [[trace norm]]. It relates to the [[negativity]] quant-ph/0102117 as follows: :$E\_N(\backslash rho)\; :=\; \backslash log\_2(\; 2\; \backslash mathcal\{N\}\; +1).$ == 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: $E\_N(\backslash rho\; \backslash otimes\; \backslash sigma)\; =\; E\_N(\backslash rho)\; \backslash cdot\; E\_N(\backslash sigma)$ * is not asymptotically continuous. That means that for a sequence of [[bipartite]] Hilbert spaces $H\_1,\; H\_2,\; \backslash ldots$ (typically with increasing dimension) we can have a sequence of quantum states $\backslash rho\_1,\; \backslash rho\_2,\; \backslash ldots$ which converges to $\backslash rho^\{\backslash otimes\; n\_1\},\; \backslash rho^\{\backslash otimes\; n\_2\},\; \backslash ldots$ (typically with increasing $n\_i$) in the [[trace distance]], but the sequence $E\_N(\backslash rho\_1)/n\_1,\; E\_N(\backslash rho\_2)/n\_2,\; \backslash ldots$ does not converge to $E\_N(\backslash rho)$. * is an upper bound to the [[distillable entanglement]] == See also == * [[Negativity]] [[Category:Quantum Information Theory]] [[Category:Handbook of Quantum Information]] [[Category:Entanglement]]