Negativity

The '''negativity'''quant-ph/0102117 is an [[entanglement measure]] which is easy to compute. The negativity can be defined as: :$\backslash mathcal\{N\}(\backslash rho)\; :=\; \backslash frac\{||\backslash rho^\{\backslash Gamma\_A\}||\_1-1\}\{2\}$ where: * $\backslash rho^\{\backslash Gamma\_A\}$ is the [[partial transpose]] of $\backslash rho$ with respect to subsystem $A$ * $||X||\_1\; =\; Tr|X|\; =\; Tr\; \backslash sqrt\{X^\backslash dagger\; X\}$ is the [[trace norm]] or the sum of the sigular values of the operator $X$. An alternative and equivalent definition is the absolute sum of the negative eigenvalues of $\backslash rho^\{\backslash Gamma\_A\}$: :$\backslash mathcal\{N\}(\backslash rho)\; :=\; \backslash sum\_i\; \backslash frac\{|\backslash lambda\_\{i\}|-\backslash lambda\_\{i\}\}\{2\}$ where $\backslash lambda\_i$ are all of the eigenvalues. '''Properties''' * Is a [[convex function]] of $\backslash rho$: :$\backslash mathcal\{N\}(\backslash sum\_\{i\}p\_\{i\}\backslash rho\_\{i\})\; \backslash le\; \backslash sum\_\{i\}p\_\{i\}\backslash mathcal\{N\}(\backslash rho\_\{i\})$ * Is an [[entanglement monotone]]: :$\backslash mathcal\{N\}(P(\backslash rho\_\{i\}))\; \backslash le\; \backslash mathcal\{N\}(\backslash rho\_\{i\})$ :where $P(\backslash rho)$ is an arbitrary [[LOCC]] operation over $\backslash rho$ == See also == * [[Logarithmic negativity]] [[Category:Quantum Information Theory]] [[Category:Handbook of Quantum Information]] [[Category:Entanglement]]