# Superfidelity

'''Superfidelity''' is a measure of similarity between density operators. It is defined as $G\left(\rho,\sigma\right) = \mathrm\left\{tr\right\}\rho\sigma + \sqrt\left\{1-\mathrm\left\{tr\right\}\left(\rho^2\right)\right\}\sqrt\left\{1-\mathrm\left\{tr\right\}\left(\sigma^2\right)\right\},$ where $\sigma$ and $\rho$ 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: $0 \le G\left(\rho_1,\rho_2\right) \le 1$. * Symmetry: $\; G\left(\rho_1,\rho_2\right)= G\left(\rho_2,\rho_1\right)$. * Unitary invariance: for any unitary operator $\ U$, we have $\; G\left(\rho_1,\rho_2\right)=G\left(U\rho_1U^\left\{\dagger\right\},U\rho_2U^\left\{\dagger\right\}\right)$. * Concavity: : $G\left(\rho_1,\alpha \rho_2 + \left(1-\alpha\right)\rho_3\right) \geq \alpha G\left(\rho_1,\rho_2\right) + \left(1-\alpha\right) G\left(\rho_1,\rho_3\right)$ for any $\rho_1,\rho_2,\rho_3\in\Omega_N$ and $\alpha \in \left[0,1\right]$. * Supermultiplicativity: for $\rho_1,\rho_2,\rho_3,\rho_4 \in \Omega_N$ we have : $G\left(\rho_1 \otimes \rho_2, \rho_3 \otimes \rho_4\right) \geq G\left(\rho_1,\rho_3\right) G\left(\rho_2,\rho_4\right).$ == See also == * Fidelity * Trace distance * Trace norm == References == Category:Handbook of Quantum Information Category:Mathematical Structure Category:Linear Algebra