# Fidelity

== Quantum Fidelity == Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties and it can be used to defined a metric on this space of density matrices, known as Bures metric. Fidelity as a distance measure between pure states used to be called "transition probability". For two states given by unit vectors $\phi,\psi$ it is $\vert\langle\phi,\psi\rangle\vert^2$. For a pure state (vector $\psi$) and a mixed state (density matrix $\rho$) this generalizes to $\langle\psi,\rho\psi\rangle$, and for two density matrices $\rho,\sigma$ it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem by Uhlmann, this leads to the expression :$F\left(\rho,\sigma\right)=\left\left(\textrm\left\{tr\right\}\sqrt\left\{\sqrt\left\{\rho\right\}\sigma\sqrt\left\{\rho\right\}\right\}\right\right)^2$ This is precisely the expression used by Richard Jozsa inJozsa94, where the term fidelity appears to have been used first. However, one can also start from $\vert\langle\phi,\psi\rangle\vert$, leading to the alternative :$F\text{'}\left(\rho,\sigma\right)=\textrm\left\{tr\right\}\sqrt\left\{\sqrt\left\{\rho\right\}\sigma\sqrt\left\{\rho\right\}\right\}$ used inNielsenChuang. This second quantity is sometimes denoted as $\sqrt\left\{F\right\}$ and called ''square root fidelity''. It has no interpretation as a probability, but appears in some estimates in a simpler way. == Basic properties == If $\rho=|\psi\rangle\langle\psi|$ is pure, then $F\left(\rho,\sigma\right)=\langle\psi|\sigma|\psi\rangle$ and if both states are pure i.e. $\rho=|\psi\rangle\langle\psi|$ and $\sigma=|\phi\rangle\langle\phi|$, then $F\left(\rho,\sigma\right)=|\langle\psi|\phi\rangle|^2$. Other properties: # $0\leq F\left(\rho,\sigma\right)\leq 1$ # $F\left(\rho,\sigma\right)=F\left(\sigma,\rho\right)\,$ # $F\left(\rho_1\otimes\rho_2,\sigma_1\otimes\sigma_2\right)=F\left(\rho_1,\sigma_1\right)F\left(\rho_2,\sigma_2\right)$ # $F\left(U \rho U^\dagger,U\sigma U^\dagger\right)=F\left(\rho,\sigma\right)$ # $F\left(\rho,\alpha\sigma_1+\left(1-\alpha\right)\sigma_2\right)\geq \alpha F\left(\rho,\sigma_1\right)+\left(1-\alpha\right)F\left(\rho,\sigma_2\right),\ \alpha\in\left[0,1\right]$ == Bures distance == Fidelity can be used to define metric on the set of quantum states, so called ''Bures distance''fuchs96phd $D_B$ :$D_B\left(\rho,\sigma\right) = \sqrt\left\{2-2\sqrt\left\{F\left(\rho,\sigma\right)\right\}\right\}$ and the ''angle''NielsenChuang :$D_A\left(\rho,\sigma\right) = \arccos\sqrt\left\{F\left(\rho,\sigma\right)\right\}.$ The quantity $D_B\left(\rho,\sigma\right)$ is the minimal distance between purifications of $\rho$ and $\sigma$ using a common environment. == Classical fidelity == Fidelity is also defined for classical probability distributions. Let $\\left\{p_i\\right\}$ and $\\left\{q_i\\right\}$ where $i=1,2,...,n$ be probability distributions. The fidelity between p and q is defined as $F\text{'}\left(\\left\{p_i\\right\},\\left\{q_i\\right\}\right)=\sum_\left\{i=1\right\}^n\sqrt\left\{p_i,q_i\right\}.$ == References == == See also == * Trace distance * Trace norm * Superfidelity Category:Handbook of Quantum Information Category:Mathematical Structure Category:Linear Algebra