Superfidelity

'''Superfidelity''' is a measure of similarity between [[density operator]]s. It is defined as $G(\backslash rho,\backslash sigma)\; =\; \backslash mathrm\{tr\}\backslash rho\backslash sigma\; +\; \backslash sqrt\{1-\backslash mathrm\{tr\}(\backslash rho^2)\}\backslash sqrt\{1-\backslash mathrm\{tr\}(\backslash sigma^2)\},$ where $\backslash sigma$ and $\backslash rho$ are density matrices. Superfidelity was introduced inmiszczak09sub as an upper bound for [[fidelity]]. == Properties == Super-fidelity has also properties which make it useful for quantifying distance between quantum states. In particular we have: * Bounds: $0\; \backslash le\; G(\backslash rho\_1,\backslash rho\_2)\; \backslash le\; 1$. * Symmetry: $\backslash ;\; G(\backslash rho\_1,\backslash rho\_2)=\; G(\backslash rho\_2,\backslash rho\_1)$. * Unitary invariance: for any unitary operator $\backslash \; U$, we have $\backslash ;\; G(\backslash rho\_1,\backslash rho\_2)=G(U\backslash rho\_1U^\{\backslash dagger\},U\backslash rho\_2U^\{\backslash dagger\})$. * Concavity: : $G(\backslash rho\_1,\backslash alpha\; \backslash rho\_2\; +\; (1-\backslash alpha)\backslash rho\_3)\; \backslash geq\; \backslash alpha\; G(\backslash rho\_1,\backslash rho\_2)\; +\; (1-\backslash alpha)\; G(\backslash rho\_1,\backslash rho\_3)$ for any $\backslash rho\_1,\backslash rho\_2,\backslash rho\_3\backslash in\backslash Omega\_N$ and $\backslash alpha\; \backslash in\; [0,1]$. * Supermultiplicativity: for $\backslash rho\_1,\backslash rho\_2,\backslash rho\_3,\backslash rho\_4\; \backslash in\; \backslash Omega\_N$ we have : $G(\backslash rho\_1\; \backslash otimes\; \backslash rho\_2,\; \backslash rho\_3\; \backslash otimes\; \backslash rho\_4)\; \backslash geq\; G(\backslash rho\_1,\backslash rho\_3)\; G(\backslash rho\_2,\backslash rho\_4).$ == See also == * [[Fidelity]] * [[Trace distance]] * [[Trace norm]] == References == [[Category:Handbook of Quantum Information]] [[Category:Mathematical Structure]] [[Category:Linear Algebra]]